National Engineering School of Brest Brittany ENIB Entrance Exam Set

The scenario describes a system where a signal is transmitted through a medium with varying impedance. The reflection coefficient at an interface is defined as \( \Gamma = \frac{Z_2 – Z_1}{Z_2 + Z_1} \), where \( Z_1 \) is the impedance of the first medium and \( Z_2 \) is the impedance of the second medium. In this case, the transmitted signal encounters an interface between a transmission line with characteristic impedance \( Z_{TL} = 50 \, \Omega \) and a load with impedance \( Z_L \). The reflection coefficient at this load is \( \Gamma_L = \frac{Z_L – Z_{TL}}{Z_L + Z_{TL}} \). The problem states that the reflected signal has an amplitude that is 20% of the incident signal’s amplitude, meaning \( |\Gamma_L| = 0.20 \). We are given that the load impedance \( Z_L \) is purely resistive, so \( Z_L = R_L \). Therefore, \( \Gamma_L = \frac{R_L – Z_{TL}}{R_L + Z_{TL}} \). Since \( |\Gamma_L| = 0.20 \), we have \( \left| \frac{R_L – 50}{R_L + 50} \right| = 0.20 \). This implies two possibilities for the real value of \( \frac{R_L – 50}{R_L + 50} \): 1) \( \frac{R_L – 50}{R_L + 50} = 0.20 \) \( R_L – 50 = 0.20 (R_L + 50) \) \( R_L – 50 = 0.20 R_L + 10 \) \( R_L – 0.20 R_L = 10 + 50 \) \( 0.80 R_L = 60 \) \( R_L = \frac{60}{0.80} = 75 \, \Omega \) 2) \( \frac{R_L – 50}{R_L + 50} = -0.20 \) \( R_L – 50 = -0.20 (R_L + 50) \) \( R_L – 50 = -0.20 R_L – 10 \) \( R_L + 0.20 R_L = 50 – 10 \) \( 1.20 R_L = 40 \) \( R_L = \frac{40}{1.20} = \frac{400}{12} = \frac{100}{3} \approx 33.33 \, \Omega \) The question asks for the load impedance that would result in a reflected signal with an amplitude 20% of the incident signal. Both \( 75 \, \Omega \) and \( \frac{100}{3} \, \Omega \) satisfy this condition. However, the context of signal integrity and impedance matching in electrical engineering, particularly at institutions like ENIB which emphasize practical applications and advanced concepts in telecommunications and electronics, often involves understanding the behavior of signals in transmission lines. A load impedance of \( 75 \, \Omega \) represents a mismatch that causes a reflection, and it’s a common value encountered in certain RF applications. The value \( \frac{100}{3} \, \Omega \) also represents a mismatch. Without further context to prefer one over the other (e.g., minimizing reflection magnitude in a specific direction or for a specific type of signal), both are mathematically valid. However, in typical problem sets designed to test understanding of reflection coefficients, a single, clear answer is usually expected. The phrasing “the load impedance” suggests a unique solution is sought. In many practical scenarios, deviations from the characteristic impedance are considered. The value \( 75 \, \Omega \) is a straightforward resistive mismatch. The value \( \frac{100}{3} \, \Omega \) is also a resistive mismatch. The question is designed to test the understanding of the reflection coefficient formula and its application to resistive loads. Both values are plausible. Given the options, we need to select the one that is presented. Let’s re-evaluate the options. The question asks for *the* load impedance. This implies a single correct answer among the choices. Both \( 75 \, \Omega \) and \( \frac{100}{3} \, \Omega \) result in a reflection coefficient magnitude of 0.2. If the question were phrased to ask for *possible* load impedances, then both would be correct. Since it asks for *the* load impedance, and we must choose one option, we consider which value might be more commonly tested or represent a distinct scenario. The value \( 75 \, \Omega \) is a common impedance value in some RF applications, and it represents a mismatch where the load impedance is higher than the characteristic impedance. The value \( \frac{100}{3} \, \Omega \) represents a mismatch where the load impedance is lower. Both are valid calculations. The question tests the fundamental understanding of the reflection coefficient. The core concept being tested is the relationship between load impedance, characteristic impedance, and the reflection coefficient magnitude. The reflection coefficient \( \Gamma \) quantifies the ratio of the reflected voltage wave to the incident voltage wave at the load. Its magnitude \( |\Gamma| \) indicates the proportion of the signal that is reflected back towards the source. A magnitude of 0.2 means 20% of the signal’s amplitude is reflected. The calculation correctly derives the two possible resistive load impedances that satisfy this condition. The question is designed to be challenging by presenting a scenario where two valid mathematical solutions exist for a resistive load, requiring the candidate to select the one provided in the options. The explanation focuses on the derivation of these values from the fundamental formula for the reflection coefficient. The correct answer is \( 75 \, \Omega \).

The scenario describes a system where a signal is transmitted through a medium and then processed by a receiver. The core concept being tested is the impact of signal degradation and noise on the fidelity of the received information, and how error detection and correction mechanisms are employed. In digital communication, the Bit Error Rate (BER) is a crucial metric. A BER of \(10^{-4}\) means that, on average, one bit in every ten thousand bits transmitted is received incorrectly. Consider a data stream of \(10^6\) bits. If the BER is \(10^{-4}\), the expected number of bit errors would be \(10^6 \times 10^{-4} = 100\) bits. The question asks about the most effective strategy to mitigate the impact of these errors, given the context of advanced engineering studies at the National Engineering School of Brest Brittany ENIB. While simply increasing transmission power might reduce some types of noise, it doesn’t inherently correct errors that have already occurred due to channel impairments or interference. Increasing the data rate without addressing error resilience would likely exacerbate the problem. The most robust approach to ensure data integrity in the face of channel noise and potential errors is through sophisticated error detection and correction coding. Techniques like Forward Error Correction (FEC) allow the receiver to detect and correct a certain number of errors without requiring retransmission. This is a fundamental principle in reliable digital communication systems, a key area of study in telecommunications and signal processing at institutions like ENIB. Therefore, implementing advanced error correction codes is the most direct and effective method to improve the reliability of the received data stream, ensuring that the intended information is accurately reconstructed despite the inherent imperfections of the transmission channel.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency of \(f_c = 10 \text{ kHz}\). The second filter is a band-pass filter with a lower cutoff frequency of \(f_{L} = 5 \text{ kHz}\) and an upper cutoff frequency of \(f_{H} = 15 \text{ kHz}\). The third filter is a high-pass filter with a cutoff frequency of \(f_{hp} = 8 \text{ kHz}\). A signal containing frequencies from \(2 \text{ kHz}\) to \(20 \text{ kHz}\) is applied to this cascaded system. We need to determine which frequencies from the original signal will pass through the entire system. 1. **First Filter (Low-Pass, \(f_c = 10 \text{ kHz}\)):** This filter allows frequencies from \(0 \text{ Hz}\) up to \(10 \text{ kHz}\) to pass. The input signal is \(2 \text{ kHz}\) to \(20 \text{ kHz}\). After the first filter, the frequencies that pass are from \(2 \text{ kHz}\) to \(10 \text{ kHz}\). 2. **Second Filter (Band-Pass, \(f_L = 5 \text{ kHz}\), \(f_H = 15 \text{ kHz}\)):** This filter allows frequencies between \(5 \text{ kHz}\) and \(15 \text{ kHz}\) to pass. The signal entering this filter has frequencies from \(2 \text{ kHz}\) to \(10 \text{ kHz}\). The intersection of these two ranges is \(5 \text{ kHz}\) to \(10 \text{ kHz}\). 3. **Third Filter (High-Pass, \(f_{hp} = 8 \text{ kHz}\)):** This filter allows frequencies above \(8 \text{ kHz}\) to pass. The signal entering this filter has frequencies from \(5 \text{ kHz}\) to \(10 \text{ kHz}\). The intersection of these two ranges is \(8 \text{ kHz}\) to \(10 \text{ kHz}\). Therefore, the frequencies that will pass through the entire cascaded system are those between \(8 \text{ kHz}\) and \(10 \text{ kHz}\). This analysis is fundamental in signal processing and telecommunications, areas of significant focus at the National Engineering School of Brest Brittany ENIB, where understanding spectral manipulation is crucial for designing communication systems and analyzing sensor data. The ability to predict the output spectrum of a cascaded filter system demonstrates a core competency in applied signal analysis, essential for engineers working with radio frequency systems, audio processing, or biomedical signal interpretation. The careful consideration of cutoff frequencies and their interaction is paramount to achieving desired signal characteristics and avoiding unwanted noise or distortion.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency of \(f_c = 10\) kHz. The second filter is a band-pass filter with a lower cutoff frequency of \(f_{L} = 5\) kHz and an upper cutoff frequency of \(f_{H} = 15\) kHz. The third component is a high-pass filter with a cutoff frequency of \(f_{hp} = 8\) kHz. A signal containing frequencies from 3 kHz to 12 kHz is input. We need to determine which frequencies from the input signal will pass through the entire system. 1. **First Filter (Low-Pass, \(f_c = 10\) kHz):** This filter allows frequencies below 10 kHz to pass. * Input frequencies: 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 11 kHz, 12 kHz. * Frequencies passing: 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz. (Frequencies at or above 10 kHz are attenuated). 2. **Second Filter (Band-Pass, \(f_L = 5\) kHz, \(f_H = 15\) kHz):** This filter allows frequencies between 5 kHz and 15 kHz (exclusive of the cutoff frequencies, or inclusive depending on filter definition, but for conceptual understanding, we consider the passband). * Frequencies from step 1: 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz. * Frequencies passing this filter: 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz. (3 kHz and 4 kHz are below the lower cutoff). 3. **Third Filter (High-Pass, \(f_{hp} = 8\) kHz):** This filter allows frequencies above 8 kHz to pass. * Frequencies from step 2: 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz. * Frequencies passing this filter: 9 kHz. (5 kHz, 6 kHz, 7 kHz, and 8 kHz are at or below the cutoff). Therefore, only the 9 kHz frequency component from the original input signal will successfully pass through all three stages of filtering. This process demonstrates the cascaded effect of filters, where the output of one filter becomes the input for the next, progressively shaping the frequency spectrum of the signal. Understanding these sequential filtering operations is crucial in signal processing applications, such as audio engineering, telecommunications, and sensor data analysis, where specific frequency bands need to be isolated or removed. The National Engineering School of Brest Brittany (ENIB) emphasizes such practical signal processing concepts in its curriculum, preparing students for real-world engineering challenges.

The question probes the understanding of the fundamental principles governing the operation of a synchronous generator, specifically focusing on the relationship between excitation current, terminal voltage, and power factor under varying load conditions. A synchronous generator’s terminal voltage is influenced by the internal generated voltage (proportional to excitation current), the synchronous reactance, and the armature resistance. When a generator supplies a lagging power factor load, the armature reaction demagnetizes the main field, leading to a drop in terminal voltage. Conversely, a leading power factor load results in a magnetizing armature reaction, which boosts the terminal voltage. To maintain a constant terminal voltage as the load changes from no-load to full-load with a lagging power factor, the excitation current must be increased. This increased excitation compensates for the voltage drop caused by the armature reaction and the synchronous reactance. The internal generated voltage, \(E_f\), which is directly related to the excitation current, needs to be sufficiently high to overcome the voltage drops across the synchronous reactance (\(jX_s\)) and armature resistance (\(R_a\)), and to account for the demagnetizing effect of the armature current (\(I_a\)) at a lagging power factor. The phasor diagram for a lagging power factor load shows that \(E_f\) must be larger than the terminal voltage \(V_t\) to maintain \(V_t\) constant. Conversely, if the load has a leading power factor, the armature reaction is magnetizing, and the excitation current might need to be decreased to keep the terminal voltage constant. At unity power factor, the armature reaction is neither significantly demagnetizing nor magnetizing, and the excitation requirement is intermediate. Therefore, to maintain a stable terminal voltage across different load power factors, particularly when transitioning from no-load to a lagging power factor load, a significant increase in excitation is typically required. This principle is a cornerstone of synchronous machine operation and is crucial for grid stability and power quality, areas of significant focus in electrical engineering programs at institutions like ENIB.

The scenario describes a system where a sensor network is deployed to monitor environmental conditions in a coastal region near Brest, a key area of focus for the National Engineering School of Brest Brittany ENIB Entrance Exam due to its maritime and technological significance. The core of the problem lies in understanding how to optimize the data transmission from these sensors to a central processing unit, considering factors like energy efficiency and data integrity, which are paramount in embedded systems and telecommunications engineering, disciplines strongly represented at ENIB. The question probes the understanding of different communication protocols and their suitability for a resource-constrained, distributed system. Let’s analyze the options in the context of ENIB’s emphasis on robust, efficient, and scalable engineering solutions. Option A: Utilizing a low-power wide-area network (LPWAN) protocol like LoRaWAN for sensor data transmission. LoRaWAN is designed for long-range, low-bandwidth communication with minimal power consumption, making it ideal for battery-powered sensors deployed over large geographical areas. Its mesh networking capabilities can enhance data reliability by providing alternative routes if a direct connection to the gateway is obstructed, a common issue in complex coastal terrains. The adaptive data rate (ADR) feature further optimizes energy usage by adjusting the transmission parameters based on the link quality. This aligns with ENIB’s focus on sustainable and efficient technological solutions. Option B: Employing a high-bandwidth, low-latency protocol such as Wi-Fi for all sensor nodes. While Wi-Fi offers high data rates, its power consumption is significantly higher than LPWAN technologies, making it unsuitable for long-term, battery-operated sensor deployments. The limited range of standard Wi-Fi would also necessitate a dense deployment of access points, increasing infrastructure costs and complexity. This approach would not be energy-efficient, a critical consideration for ENIB’s practical engineering applications. Option C: Relying solely on Bluetooth Low Energy (BLE) for direct sensor-to-gateway communication. BLE is designed for short-range, low-power communication, typically between devices within a few meters. Its range is insufficient for covering a wide coastal area, and it lacks the robust mesh networking capabilities required for reliable data transmission in potentially challenging environments. While power-efficient for short distances, its limited scope makes it impractical for this scenario. Option D: Implementing a traditional cellular (e.g., 4G/5G) connection for each individual sensor. While cellular networks offer wide coverage, the power consumption associated with establishing and maintaining a cellular connection for each sensor would be prohibitively high for a battery-powered sensor network. Furthermore, the cost per sensor for data plans would be substantial. This approach is generally reserved for higher-bandwidth applications or when cellular infrastructure is readily available and power is not a primary constraint. Therefore, LoRaWAN (Option A) represents the most appropriate and efficient solution for the described sensor network, reflecting ENIB’s commitment to innovative and resource-conscious engineering practices.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of data acquisition and processing, a fundamental concept in many engineering disciplines taught at ENIB, particularly in areas like signal processing, telecommunications, and instrumentation. While no explicit calculation is required, the underlying concept involves how noise impacts the discernibility of a true signal. Signal-to-noise ratio (SNR) is defined as the ratio of the power of a signal to the power of background noise. A higher SNR indicates that the signal is clearer relative to the noise. In the context of data acquisition, increasing the sampling rate without a corresponding increase in the signal’s bandwidth or the analog-to-digital converter’s (ADC) resolution will not inherently improve the SNR. Instead, it might even introduce more quantization noise if the ADC’s precision is exceeded by the finer steps. To improve SNR, one typically needs to either increase the signal power (e.g., by using a stronger transmitter or a more sensitive sensor), reduce the noise power (e.g., by using better shielding, filtering, or cooling), or employ signal processing techniques like averaging multiple measurements (which reduces random noise by a factor of the square root of the number of averages, assuming the signal is coherent). Considering the options: 1. Increasing the sampling rate alone: This primarily affects the frequency resolution and the ability to capture high-frequency components of the signal. It does not directly increase the power of the signal relative to the noise floor unless the original sampling rate was too low to accurately represent the signal’s bandwidth, leading to aliasing. 2. Reducing the analog signal’s amplitude: This would decrease the signal power, thus *decreasing* the SNR. 3. Averaging multiple independent measurements of the same signal under identical conditions: This is a standard technique to improve SNR. Random noise, by its nature, tends to average out over multiple trials, while the coherent signal remains consistent. If \(N\) independent measurements are averaged, the noise power is reduced by a factor of \(N\), and the noise amplitude by a factor of \(\sqrt{N}\), thereby increasing the SNR by \(\sqrt{N}\). 4. Increasing the bit depth of the analog-to-digital converter (ADC) without changing the sampling rate: Increasing bit depth increases the dynamic range and reduces quantization noise, which is a form of noise inherent in the digitization process. This directly improves the SNR by providing finer steps to represent the analog signal. Therefore, both increasing bit depth and averaging measurements are effective methods for improving SNR. The question asks for a method that *enhances* the signal’s clarity relative to noise. Averaging multiple measurements is a robust and widely applicable technique for this purpose, directly addressing the statistical nature of random noise. Increasing bit depth also directly improves SNR by reducing quantization error. However, averaging is often considered a more fundamental way to combat additive, random noise in many signal acquisition scenarios, especially when the signal source is stable. The question asks for a method that “enhances the clarity of the signal relative to background noise.” Averaging directly reduces the impact of random noise on the overall measurement. Let’s re-evaluate the options in terms of direct impact on SNR. Increasing bit depth of the ADC directly reduces quantization noise, which is a component of the overall noise floor. If the original bit depth was insufficient, this would be a primary way to improve SNR. Averaging multiple measurements reduces the impact of random noise. The question is phrased to test a nuanced understanding of SNR improvement techniques. Both increasing bit depth and averaging are valid. However, averaging is a technique that works by exploiting the statistical properties of noise and signal over time, effectively “pulling out” the signal from a noisy background. This is a core concept in many signal processing applications at ENIB. Let’s assume the scenario implies that the signal is stable enough to be measured repeatedly. In this context, averaging is a very direct method to improve SNR. If the ADC’s bit depth is already sufficient for the signal’s dynamic range, increasing it further might yield diminishing returns or be limited by other noise sources. Averaging, however, consistently improves SNR by \(\sqrt{N}\) for \(N\) averages, provided the noise is random and uncorrelated between measurements. The most universally applicable and conceptually distinct method for improving SNR by reducing the impact of random noise, without necessarily altering the signal itself or the fundamental digitization process (beyond its initial setup), is averaging. Final Answer is averaging.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency \(f_c = 1000 \text{ Hz}\). The second filter is a band-pass filter with a lower cutoff frequency \(f_{L} = 500 \text{ Hz}\) and an upper cutoff frequency \(f_{H} = 1500 \text{ Hz}\). The third filter is a high-pass filter with a cutoff frequency \(f_{hp} = 750 \text{ Hz}\). A signal containing frequencies from \(200 \text{ Hz}\) to \(1200 \text{ Hz}\) is applied to this cascaded system. We need to determine which frequencies from the original signal will pass through all three filters. 1. **First Filter (Low-Pass, \(f_c = 1000 \text{ Hz}\)):** This filter allows frequencies below \(1000 \text{ Hz}\) to pass. Applying this to the original signal (\(200 \text{ Hz}\) to \(1200 \text{ Hz}\)), the output will contain frequencies from \(200 \text{ Hz}\) to \(1000 \text{ Hz}\). 2. **Second Filter (Band-Pass, \(f_L = 500 \text{ Hz}\), \(f_H = 1500 \text{ Hz}\)):** This filter allows frequencies between \(500 \text{ Hz}\) and \(1500 \text{ Hz}\) to pass. When applied to the output of the first filter (\(200 \text{ Hz}\) to \(1000 \text{ Hz}\)), the frequencies that are common to both ranges are from \(500 \text{ Hz}\) to \(1000 \text{ Hz}\). 3. **Third Filter (High-Pass, \(f_{hp} = 750 \text{ Hz}\)):** This filter allows frequencies above \(750 \text{ Hz}\) to pass. When applied to the output of the second filter (\(500 \text{ Hz}\) to \(1000 \text{ Hz}\)), the frequencies that are common to both ranges are from \(750 \text{ Hz}\) to \(1000 \text{ Hz}\). Therefore, the frequencies that will successfully pass through all three cascaded filters are those between \(750 \text{ Hz}\) and \(1000 \text{ Hz}\). This type of analysis is fundamental in signal processing and communications engineering, areas of significant focus at the National Engineering School of Brest Brittany ENIB, where understanding spectral manipulation is crucial for designing robust communication systems and analyzing sensor data. The ability to predict the output spectrum of a cascaded filter system demonstrates a core competency in applied signal analysis, essential for students pursuing advanced studies in fields like telecommunications, embedded systems, and signal intelligence. The sequential application of filter characteristics highlights the importance of understanding the cumulative effect of system components, a principle that extends beyond signal processing to areas like control systems and circuit design, both integral to the ENIB curriculum.

The question probes the understanding of signal processing concepts, specifically the Nyquist-Shannon sampling theorem and its practical implications in digital signal reconstruction. The theorem states that to perfectly reconstruct a continuous-time signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In this scenario, the analog signal has a maximum frequency component of 15 kHz. Therefore, the minimum sampling frequency required to avoid aliasing and allow for perfect reconstruction is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the consequence of sampling at a frequency *below* this Nyquist rate. When sampling below the Nyquist rate, higher frequency components in the analog signal are misrepresented as lower frequencies in the sampled digital signal. This phenomenon is called aliasing. Aliasing distorts the reconstructed signal, making it appear as if it contains frequencies that were not originally present or altering the perceived frequencies of the original components. Specifically, frequencies above \(f_s/2\) (the Nyquist frequency) will fold back into the range \(0\) to \(f_s/2\). Consider a frequency \(f > f_s/2\). The aliased frequency \(f_{alias}\) is given by \(f_{alias} = |f – k \cdot f_s|\), where \(k\) is an integer chosen such that \(0 \le f_{alias} < f_s/2\). If we sample at \(f_s = 20 \text{ kHz}\) and the signal contains a component at 15 kHz, this 15 kHz component is above \(f_s/2 = 10 \text{ kHz}\). Using the aliasing formula with \(k=1\), \(f_{alias} = |15 \text{ kHz} – 1 \cdot 20 \text{ kHz}| = |-5 \text{ kHz}| = 5 \text{ kHz}\). Thus, the 15 kHz component will be incorrectly represented as a 5 kHz component in the sampled data. This demonstrates that the reconstruction will be inaccurate, as the original 15 kHz information is lost and replaced by a spurious 5 kHz signal. The ability to accurately reconstruct the original signal is compromised due to the violation of the sampling theorem.

The scenario describes a system where a signal is transmitted and received. The core concept being tested is the impact of signal attenuation and noise on the signal-to-noise ratio (SNR) at the receiver. Signal power at the transmitter \(P_{tx}\) is given as 10 dBm. Attenuation factor \(L\) is 15 dB. Noise power spectral density \(N_0\) is \(-174\) dBm/Hz. Bandwidth \(B\) is 1 MHz, which is \(10^6\) Hz. First, convert the transmitter power from dBm to Watts: \(P_{tx\_W} = 10^{\frac{P_{tx\_dBm}}{10}} \times 10^{-3}\) \(P_{tx\_W} = 10^{\frac{10}{10}} \times 10^{-3} = 10^1 \times 10^{-3} = 0.01\) Watts. Next, calculate the received signal power \(P_{rx}\) in dBm. Attenuation is a loss, so it’s subtracted: \(P_{rx\_dBm} = P_{tx\_dBm} – L\) \(P_{rx\_dBm} = 10 \text{ dBm} – 15 \text{ dB} = -5 \text{ dBm}\). Now, convert the received signal power to Watts: \(P_{rx\_W} = 10^{\frac{P_{rx\_dBm}}{10}} \times 10^{-3}\) \(P_{rx\_W} = 10^{\frac{-5}{10}} \times 10^{-3} = 10^{-0.5} \times 10^{-3} \approx 0.3162 \times 10^{-3} = 0.0003162\) Watts. Calculate the total noise power \(P_n\) in Watts. First, convert noise power spectral density to Watts/Hz: \(N_{0\_W/Hz} = 10^{\frac{N_{0\_dBm/Hz}}{10}} \times 10^{-3}\) \(N_{0\_W/Hz} = 10^{\frac{-174}{10}} \times 10^{-3} = 10^{-17.4} \times 10^{-3} \approx 3.981 \times 10^{-21} \times 10^{-3} = 3.981 \times 10^{-24}\) W/Hz. Total noise power is \(P_n = N_0 \times B\): \(P_n = (3.981 \times 10^{-24} \text{ W/Hz}) \times (10^6 \text{ Hz})\) \(P_n = 3.981 \times 10^{-18}\) Watts. Now, calculate the Signal-to-Noise Ratio (SNR) in Watts: \(SNR_W = \frac{P_{rx\_W}}{P_n}\) \(SNR_W = \frac{0.0003162 \text{ W}}{3.981 \times 10^{-18} \text{ W}} \approx 7.94 \times 10^{13}\). Finally, convert the SNR to decibels (dB): \(SNR_{dB} = 10 \log_{10}(SNR_W)\) \(SNR_{dB} = 10 \log_{10}(7.94 \times 10^{13})\) \(SNR_{dB} = 10 \times (\log_{10}(7.94) + \log_{10}(10^{13}))\) \(SNR_{dB} = 10 \times (0.90 + 13)\) \(SNR_{dB} = 10 \times 13.90 \approx 139 \text{ dB}\). The question probes the understanding of fundamental concepts in telecommunications engineering, particularly signal propagation and noise impact, which are crucial for students at the National Engineering School of Brest Brittany (ENIB) specializing in fields like embedded systems, networks, and signal processing. The calculation involves converting between linear power units (Watts) and logarithmic decibel units (dBm, dB), and understanding how attenuation and noise power spectral density affect the received signal quality. A high SNR indicates a robust communication link, essential for reliable data transmission in various applications studied at ENIB, such as wireless sensor networks or high-speed data links. The ability to accurately calculate and interpret SNR is a foundational skill for analyzing system performance and designing efficient communication protocols. The context of ENIB’s research strengths in areas like maritime technologies and intelligent systems means understanding signal integrity in potentially challenging environments is paramount.

The scenario describes a system where a signal is processed through a series of filters. The first filter has a frequency response \(H_1(j\omega) = \frac{1}{1 + j\omega}\). The second filter has a frequency response \(H_2(j\omega) = \frac{j\omega}{1 + j\omega}\). When two filters are cascaded, their frequency responses multiply. Therefore, the overall frequency response of the cascaded system is \(H_{total}(j\omega) = H_1(j\omega) \cdot H_2(j\omega)\). \(H_{total}(j\omega) = \left(\frac{1}{1 + j\omega}\right) \cdot \left(\frac{j\omega}{1 + j\omega}\right) = \frac{j\omega}{(1 + j\omega)^2}\) To analyze the phase response, we need to find the argument of \(H_{total}(j\omega)\). The numerator is \(j\omega\), which has an argument of \(\frac{\pi}{2}\) for \(\omega > 0\). The denominator is \((1 + j\omega)^2\). The argument of \(1 + j\omega\) is \(\arctan(\omega)\). Therefore, the argument of \((1 + j\omega)^2\) is \(2 \cdot \arctan(\omega)\). The phase of \(H_{total}(j\omega)\) is the argument of the numerator minus the argument of the denominator: \(\phi(\omega) = \arg(j\omega) – \arg((1 + j\omega)^2)\) \(\phi(\omega) = \frac{\pi}{2} – 2 \arctan(\omega)\) Now, let’s evaluate the phase at a specific frequency, for instance, \(\omega = 1\) rad/s. \(\phi(1) = \frac{\pi}{2} – 2 \arctan(1)\) Since \(\arctan(1) = \frac{\pi}{4}\), \(\phi(1) = \frac{\pi}{2} – 2 \left(\frac{\pi}{4}\right) = \frac{\pi}{2} – \frac{\pi}{2} = 0\) radians. This means that at a frequency of 1 rad/s, the phase shift introduced by the cascaded system is 0 radians. This implies that the output signal at this frequency will be in phase with the input signal, despite the presence of two distinct filters. Understanding phase response is crucial in signal processing and control systems, areas of study at ENIB, as it dictates how different frequency components of a signal are delayed or advanced, affecting the overall signal shape and system stability. The ability to analyze and predict phase shifts in cascaded systems is a fundamental skill for engineers working with complex signal chains or feedback loops.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency \(f_c = 10 \text{ kHz}\). The second filter is a band-pass filter with a lower cutoff frequency \(f_{L} = 5 \text{ kHz}\) and an upper cutoff frequency \(f_{H} = 15 \text{ kHz}\). The third filter is a high-pass filter with a cutoff frequency \(f_{hp} = 8 \text{ kHz}\). We need to determine which frequency components from an input signal, containing frequencies at 3 kHz, 7 kHz, 12 kHz, and 18 kHz, will pass through the entire cascade of filters. 1. **First Filter (Low-Pass, \(f_c = 10 \text{ kHz}\)):** * 3 kHz: Passes (since \(3 \text{ kHz} < 10 \text{ kHz}\)) * 7 kHz: Passes (since \(7 \text{ kHz} < 10 \text{ kHz}\)) * 12 kHz: Attenuated (since \(12 \text{ kHz} > 10 \text{ kHz}\)) * 18 kHz: Attenuated (since \(18 \text{ kHz} > 10 \text{ kHz}\)) Frequencies remaining after the first filter: 3 kHz, 7 kHz. 2. **Second Filter (Band-Pass, \(f_{L} = 5 \text{ kHz}\), \(f_{H} = 15 \text{ kHz}\)):** * 3 kHz: Attenuated (since \(3 \text{ kHz} < 5 \text{ kHz}\)) * 7 kHz: Passes (since \(5 \text{ kHz} < 7 \text{ kHz} < 15 \text{ kHz}\)) Frequencies remaining after the second filter: 7 kHz. 3. **Third Filter (High-Pass, \(f_{hp} = 8 \text{ kHz}\)):** * 7 kHz: Attenuated (since \(7 \text{ kHz} < 8 \text{ kHz}\)) Frequencies remaining after the third filter: None. Therefore, no frequency components from the original signal will pass through all three filters. The correct answer is that no frequencies will be transmitted. This question assesses understanding of cascaded filter responses, a fundamental concept in signal processing and telecommunications, areas of significant focus within the National Engineering School of Brest Brittany ENIB's curriculum, particularly in its electronics and digital systems engineering programs. Students are expected to grasp how the passband and stopband characteristics of individual filters combine to define the overall system's frequency response. Analyzing the sequential impact of each filter on a given set of frequencies requires a systematic approach, mirroring the problem-solving methodologies emphasized at ENIB. The ability to predict the output of such a system is crucial for designing communication systems, control systems, and audio processing applications, all of which are relevant to the advanced engineering disciplines taught at the institution. The question probes the practical application of filter theory, moving beyond simple definitions to evaluate a candidate's ability to synthesize knowledge about filter interactions.

The core concept here relates to the principles of signal processing and information theory, specifically concerning the Nyquist-Shannon sampling theorem and its implications for digital signal reconstruction. The theorem states that to perfectly reconstruct a signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal. This minimum sampling rate is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In the given scenario, a sensor is designed to capture atmospheric pressure variations, which are inherently analog signals. The system’s analog-to-digital converter (ADC) samples this pressure data at a rate of \(f_{sampling} = 100\) Hz. The question asks about the maximum frequency component that can be accurately represented and reconstructed from these samples. According to the Nyquist-Shannon sampling theorem, the highest frequency that can be unambiguously represented is half the sampling frequency. Therefore, the maximum representable frequency is \(f_{max\_representable} = \frac{f_{sampling}}{2}\). Substituting the given sampling frequency: \(f_{max\_representable} = \frac{100 \text{ Hz}}{2} = 50 \text{ Hz}\) This means that any atmospheric pressure variations occurring at frequencies above 50 Hz will be aliased, meaning they will be misrepresented as lower frequencies during the sampling process, leading to distortion and loss of information. The ability to accurately reconstruct the original analog signal from its digital samples is directly dependent on adhering to this sampling rate. Therefore, the system can accurately represent atmospheric pressure variations up to a frequency of 50 Hz. This principle is fundamental in fields like environmental monitoring and control systems, areas of interest within the engineering disciplines at the National Engineering School of Brest Brittany ENIB. Understanding these limits is crucial for designing effective data acquisition systems that avoid information loss and ensure the integrity of collected data for analysis and decision-making.

The scenario describes a system where a signal is transmitted through a medium and then processed by a receiver. The core concept being tested is the impact of signal degradation and noise on the fidelity of information. In the context of ENIB’s engineering programs, particularly those involving signal processing, telecommunications, or embedded systems, understanding the trade-offs between signal strength, noise levels, and the effectiveness of filtering is crucial. The signal-to-noise ratio (SNR) is a fundamental metric that quantifies the level of a desired signal relative to the level of background noise. A higher SNR indicates a cleaner signal with less interference, leading to more accurate reception and processing. Conversely, a lower SNR implies that the noise is more significant compared to the signal, making it harder to extract the original information. When a signal is transmitted through a medium, it can be attenuated (weakened) and corrupted by various forms of noise. Noise can originate from the transmission medium itself (e.g., thermal noise, atmospheric interference) or from the electronic components within the transmitter and receiver. The receiver often employs filtering techniques to reduce the impact of noise and isolate the desired signal. However, the effectiveness of these filters is limited, and aggressive filtering can sometimes distort the signal itself, a phenomenon known as signal distortion or bandwidth limitation. In this specific scenario, the received signal has a power of \(P_{signal} = 10^{-10}\) Watts and the noise power is \(P_{noise} = 10^{-12}\) Watts. The SNR in decibels (dB) is calculated as: \[ SNR_{dB} = 10 \log_{10} \left( \frac{P_{signal}}{P_{noise}} \right) \] \[ SNR_{dB} = 10 \log_{10} \left( \frac{10^{-10} \text{ W}}{10^{-12} \text{ W}} \right) \] \[ SNR_{dB} = 10 \log_{10} (10^2) \] \[ SNR_{dB} = 10 \times 2 \] \[ SNR_{dB} = 20 \text{ dB} \] This calculated SNR of 20 dB indicates a reasonably strong signal relative to the noise. However, the question asks about the *primary challenge* for the receiver’s signal processing unit. While a 20 dB SNR is not extremely low, the presence of noise, however small, necessitates sophisticated processing. The receiver must distinguish the desired signal from the superimposed noise. This requires algorithms that can effectively estimate the original signal’s characteristics and remove or mitigate the noise components. Therefore, the fundamental task for the receiver’s signal processing unit is to accurately reconstruct the original signal from the noisy received waveform. This involves techniques like matched filtering, adaptive filtering, or more advanced signal reconstruction algorithms, all aimed at maximizing the extraction of the signal’s information content despite the presence of noise. The challenge isn’t just the SNR value itself, but the inherent difficulty in separating a signal from any amount of noise, a core problem in signal processing and communications engineering, which are vital disciplines at ENIB.

The scenario describes a system where a signal is processed through a series of filters. The first filter is a low-pass filter with a cutoff frequency \(f_c = 1000 \text{ Hz}\). The second filter is a high-pass filter with a cutoff frequency \(f_o = 500 \text{ Hz}\). The question asks about the overall effect of cascading these two filters. A low-pass filter allows frequencies below its cutoff to pass while attenuating frequencies above it. Conversely, a high-pass filter allows frequencies above its cutoff to pass while attenuating frequencies below it. When a low-pass filter with a cutoff of 1000 Hz is followed by a high-pass filter with a cutoff of 500 Hz, the system will pass frequencies that are both below 1000 Hz (due to the low-pass filter) and above 500 Hz (due to the high-pass filter). Therefore, the combined effect is to pass frequencies within the range of 500 Hz to 1000 Hz. This type of filter configuration is known as a band-pass filter. The specific characteristic of this band-pass filter is that it allows frequencies between the lower cutoff of the high-pass filter and the upper cutoff of the low-pass filter to pass through. The National Engineering School of Brest Brittany (ENIB) often emphasizes signal processing and control systems, where understanding filter cascading is fundamental. This question tests the ability to synthesize the behavior of individual filters to predict the behavior of a composite system, a core skill in electrical engineering and related fields taught at ENIB. The correct answer is the description of a band-pass filter with the specified frequency range.

The question probes the understanding of fundamental principles in embedded systems design, specifically concerning real-time operating systems (RTOS) and interrupt handling, which are core to many engineering disciplines at ENIB. The scenario describes a critical task in a maritime navigation system, a field where ENIB has significant research interests. The core issue is how to ensure a high-priority interrupt, triggered by a GPS signal update, does not unduly delay a lower-priority but time-sensitive task responsible for collision avoidance. In an RTOS environment, interrupts are serviced by Interrupt Service Routines (ISRs). ISRs should be kept as short as possible to minimize the time the system is unresponsive to other events or tasks. When an ISR needs to communicate data or signal a task, it typically uses inter-task communication mechanisms provided by the RTOS, such as semaphores, queues, or event flags. Directly blocking a task within an ISR is generally considered bad practice as it can lead to priority inversion or deadlock. The GPS update interrupt needs to signal the navigation task that new position data is available. The collision avoidance task, being time-sensitive, needs to be able to run promptly when its conditions are met. If the GPS ISR were to directly suspend the navigation task and wait for it to process the data, it could lead to a situation where the navigation task, while processing the GPS data, is preempted by the collision avoidance task, but the GPS ISR is still holding a resource or blocking the navigation task. This creates a dependency chain that can violate real-time guarantees. The most robust approach is for the GPS ISR to signal the availability of new data to the navigation task using a non-blocking mechanism. A semaphore or a message queue is ideal for this. The ISR would signal the semaphore or place data in the queue. The navigation task, upon being woken up by the RTOS scheduler due to the signaled semaphore or queued data, would then retrieve the GPS data and process it. This decouples the ISR from the task’s execution flow, allowing the ISR to return quickly and the scheduler to manage task execution based on priorities. The collision avoidance task, if it has a higher priority than the navigation task, can then preempt the navigation task if necessary, ensuring safety. Therefore, the strategy that best addresses the real-time constraints and avoids potential deadlocks or priority inversions involves the ISR signaling the navigation task without blocking, allowing the RTOS scheduler to manage task execution efficiently. This aligns with the principles of efficient interrupt handling and task synchronization in real-time embedded systems, which are crucial for applications like those developed at ENIB.

The scenario describes a system where a signal is transmitted and received. The core concept being tested is the understanding of signal integrity and potential sources of degradation in a communication channel, particularly relevant to the fields of telecommunications and embedded systems studied at ENIB. A signal’s fidelity is compromised by various factors. Noise, which is any unwanted disturbance that interferes with the signal, is a primary concern. This can manifest as thermal noise (Johnson-Nyquist noise) due to the random motion of electrons in conductors, or external interference from electromagnetic sources. Bandwidth limitations of the transmission medium or processing components can also distort the signal by attenuating or phase-shifting certain frequency components, leading to intersymbol interference (ISI) if the signal’s components spread into adjacent symbol periods. Signal attenuation, the reduction in signal amplitude over distance or through components, weakens the signal, making it more susceptible to noise. Finally, non-linearities in amplifiers or other components can introduce harmonic distortion, creating new frequency components not present in the original signal. Considering these factors, the most encompassing and fundamental challenge to maintaining signal integrity in a complex electronic system, such as those designed and researched at ENIB, is the cumulative effect of these degradations. While specific issues like jitter or reflections are critical, the question asks for the primary challenge to the *faithful reproduction* of the original signal’s information content. Noise directly corrupts the signal’s amplitude, making it difficult to distinguish between valid states. Bandwidth limitations and attenuation alter the signal’s shape and strength, respectively, both contributing to potential misinterpretation at the receiver. Non-linearities introduce spurious signals. However, the pervasive nature of noise, coupled with its direct impact on the signal-to-noise ratio (SNR), which dictates the maximum achievable data rate and error probability, makes it the most fundamental and persistent challenge to achieving faithful signal reproduction. Without adequate noise mitigation, the impact of other impairments becomes significantly amplified. Therefore, managing and minimizing the impact of noise is paramount for ensuring the integrity of transmitted information.

The scenario describes a system where a sensor network is deployed to monitor environmental conditions in a coastal region near Brest, a key area of focus for the National Engineering School of Brest Brittany ENIB. The sensor nodes are powered by energy harvesting mechanisms, specifically solar and wave energy converters, which are crucial for sustainable operation in such maritime environments. The network’s efficiency is directly tied to the energy management strategy. The question probes the understanding of how different energy management techniques impact the network’s longevity and data acquisition capabilities. The core concept here is the trade-off between energy expenditure (for sensing, processing, and communication) and energy availability (from harvesting). A strategy that prioritizes continuous data transmission, even with intermittent harvesting, would quickly deplete battery reserves during periods of low energy generation. Conversely, a strategy that conserves energy by reducing transmission frequency or processing load during low-harvesting periods, while maximizing data capture and storage for later transmission when energy is abundant, would lead to greater overall network operational time and more comprehensive data collection over the long term. The National Engineering School of Brest Brittany ENIB emphasizes research in areas like embedded systems, renewable energy, and intelligent networks, making energy-efficient design and operation a fundamental principle. Therefore, the most effective strategy would be one that dynamically adapts to the harvested energy levels, ensuring that critical data is not lost and that the network remains functional for the longest possible duration. This involves intelligent scheduling of tasks, potentially using predictive models of energy availability, and employing low-power communication protocols. The ability to balance immediate data needs with long-term network sustainability is paramount.

The scenario describes a system where a sensor network is deployed to monitor environmental conditions in a coastal region near Brest, a key area of interest for the National Engineering School of Brest Brittany ENIB. The sensor nodes have limited battery life and processing power, necessitating efficient data aggregation and transmission strategies. The core challenge is to minimize the total energy consumed by the network while ensuring that critical data reaches a central base station within a specified latency. Consider a simplified model where \(N\) sensor nodes are distributed within a circular area of radius \(R\). Each node \(i\) has an initial energy \(E_i\) and generates data at a rate \(d_i\) bits per second. The base station is located at the center of the circle. The energy consumption for transmitting \(k\) bits of data from a node at distance \(r\) to another node or the base station is given by \(E_{tx} = k \cdot P_{tx} \cdot t_{tx}\), where \(P_{tx}\) is the transmission power and \(t_{tx}\) is the transmission time. The transmission time is proportional to the amount of data and inversely proportional to the data rate, \(t_{tx} = k / \text{data\_rate}\). The transmission power \(P_{tx}\) is typically proportional to the square of the distance, \(P_{tx} \propto r^2\). Therefore, the energy consumed for transmission is proportional to \(k \cdot r^2\). For reception, the energy consumption \(E_{rx}\) is generally less dependent on distance and is primarily related to the circuitry. The question asks about the most energy-efficient strategy for data dissemination in such a network, considering the constraints of limited resources and the need for timely data delivery. This aligns with research areas at ENIB focusing on wireless sensor networks, embedded systems, and telecommunications, particularly in maritime and environmental monitoring applications. The most energy-efficient strategy in a wireless sensor network with limited resources, especially when nodes are mobile or have varying distances to the base station, is often **cluster-based data aggregation**. In this approach, nodes are organized into clusters, with each cluster having a cluster head. Nodes transmit their data to their respective cluster heads, which then aggregate the data from all nodes in the cluster. The cluster heads then transmit the aggregated data to the base station. This strategy significantly reduces the overall energy consumption because: 1. **Reduced Transmission Distance:** Nodes transmit to their cluster head, which is typically closer than the base station, thus lowering the \(r^2\) factor in transmission energy. 2. **Data Aggregation:** Cluster heads combine data from multiple nodes, reducing the total amount of data that needs to be transmitted to the base station. This is crucial as transmission energy is often the dominant factor in sensor node power consumption. 3. **Load Balancing:** By distributing the role of cluster head among nodes over time, the energy burden is shared, preventing premature battery depletion of specific nodes. 4. **Optimized Routing:** Cluster heads can employ more sophisticated routing protocols to find the most energy-efficient path to the base station for the aggregated data. Other strategies, such as direct transmission from each node, would require higher transmission power for nodes farther from the base station, leading to rapid battery depletion. Flooding the network with data would also be highly inefficient. While direct transmission to the base station might be efficient for nodes very close to it, it is not a scalable or globally optimal solution for a network with distributed nodes and limited energy. Therefore, the strategy that best balances reduced transmission distances and data reduction through aggregation, while also considering load balancing and optimized routing, is cluster-based data aggregation. This approach is a cornerstone of energy-efficient wireless sensor network design, a topic highly relevant to the research and educational focus at the National Engineering School of Brest Brittany ENIB.

The question probes the understanding of signal processing concepts, specifically focusing on the trade-offs between time-domain and frequency-domain representations and their implications for analyzing transient signals. A fundamental principle in signal analysis is the Heisenberg Uncertainty Principle, which, when applied to signals, states that a signal cannot be simultaneously localized with arbitrary precision in both time and frequency. This means that a signal with a very short duration (highly localized in time) will necessarily have a broad frequency spectrum (poorly localized in frequency), and vice-versa. When analyzing a signal that exhibits rapid changes or transient behavior, such as a sudden impulse or a rapidly varying modulation, it is crucial to capture these temporal details accurately. Techniques that provide high temporal resolution, like the Short-Time Fourier Transform (STFT) with a short analysis window, are designed for this purpose. However, a shorter analysis window in the time domain inherently leads to a wider frequency resolution in the frequency domain. This is because the Fourier Transform of a short windowed signal will have a broader spread of frequency components. Conversely, a long analysis window would provide better frequency resolution but would blur or miss the fine temporal details of the transient event. Therefore, to effectively analyze a signal with significant transient characteristics, one must prioritize temporal resolution, accepting the inherent compromise in frequency resolution. This aligns with the core principles of signal analysis taught at institutions like the National Engineering School of Brest Brittany ENIB, where understanding these fundamental trade-offs is essential for designing and interpreting signal processing systems.

The question assesses understanding of the fundamental principles of signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its practical implications in digital signal reconstruction. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a signal from its samples, the sampling frequency \(f_s\) must be at least twice the highest frequency component \(f_{max}\) present in the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In this scenario, the analog signal contains frequency components up to \(15\) kHz. Therefore, the maximum frequency present is \(f_{max} = 15\) kHz. According to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and ensure perfect reconstruction is \(f_s \ge 2 \times f_{max}\). Calculating the minimum required sampling frequency: \(f_{s,min} = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks for the most appropriate sampling frequency for a system designed at the National Engineering School of Brest Brittany ENIB, implying a need for robust and reliable signal acquisition. While \(30\) kHz is the theoretical minimum, practical systems often employ a sampling rate slightly higher than the Nyquist rate to provide a margin of safety against imperfections in anti-aliasing filters and to simplify filter design. This margin is often referred to as “oversampling.” Considering the options: * \(20\) kHz is below the Nyquist rate (\(2 \times 15 = 30\) kHz), so it would lead to aliasing and distortion. * \(30\) kHz is the theoretical Nyquist rate. While technically sufficient, it might be too close to the limit for practical implementation without significant filter design challenges. * \(40\) kHz is above the Nyquist rate and provides a reasonable margin for practical implementation, allowing for less steep anti-aliasing filters, which are easier and cheaper to design and implement. This is a common practice in digital signal processing to ensure high-fidelity reconstruction. * \(60\) kHz is significantly higher than the Nyquist rate. While it would also avoid aliasing, it results in a higher data rate and increased computational load without a proportional increase in signal fidelity for this specific signal bandwidth. It might be considered oversampling for this particular signal, but \(40\) kHz offers a better balance between fidelity and resource utilization for a typical engineering application at ENIB. Therefore, \(40\) kHz represents the most practical and robust sampling frequency for this analog signal, balancing the theoretical requirement with real-world engineering considerations for signal reconstruction.

The question probes the understanding of signal processing, specifically the concept of aliasing and its prevention in digital signal acquisition, a core area within the curriculum of the National Engineering School of Brest Brittany (ENIB). To prevent aliasing when sampling a continuous-time signal \(x(t)\) with a maximum frequency component \(f_{max}\), the sampling frequency \(f_s\) must be at least twice the maximum frequency component. This is known as the Nyquist-Shannon sampling theorem. The minimum required sampling frequency is therefore \(f_{s,min} = 2 \times f_{max}\). In this scenario, the signal contains frequency components up to \(f_{max} = 15\) kHz. Therefore, the minimum sampling frequency required to avoid aliasing is: \(f_{s,min} = 2 \times 15 \text{ kHz} = 30 \text{ kHz}\). If the sampling frequency used is \(f_s = 25\) kHz, which is less than the minimum required \(30\) kHz, aliasing will occur. Aliasing causes higher frequency components to appear as lower frequencies in the sampled signal. Specifically, frequencies above \(f_s/2\) will be folded back into the range \(0\) to \(f_s/2\). The folding frequency, or Nyquist frequency, is \(f_N = f_s/2\). In this case, \(f_N = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\). A frequency component at \(f_{actual}\) will be aliased to \(f_{aliased}\) where \(f_{aliased} = |f_{actual} – k \cdot f_s|\) for some integer \(k\) such that \(0 \le f_{aliased} \le f_N\). Consider the highest frequency component in the original signal, which is \(15\) kHz. We need to find the aliased frequency of \(15\) kHz when sampled at \(25\) kHz. Using the formula, we look for an integer \(k\) such that \(|15 \text{ kHz} – k \times 25 \text{ kHz}|\) is within the range \(0\) to \(12.5\) kHz. For \(k=1\): \(|15 \text{ kHz} – 1 \times 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). Since \(10 \text{ kHz}\) is within the range \(0\) to \(12.5 \text{ kHz}\), the \(15\) kHz component will appear as \(10\) kHz in the sampled signal. The question asks about the consequence of sampling at \(25\) kHz when the signal has components up to \(15\) kHz. The most direct consequence of sampling below the Nyquist rate is the introduction of aliasing. The highest frequency component, \(15\) kHz, will be misrepresented. The aliased frequency is calculated as \(|15 \text{ kHz} – 1 \times 25 \text{ kHz}| = 10 \text{ kHz}\). This means that the \(15\) kHz signal will be indistinguishable from a \(10\) kHz signal after sampling. This distortion fundamentally compromises the integrity of the digital representation of the original analog signal, a critical consideration in any digital signal processing application at ENIB. Understanding and mitigating aliasing is paramount for accurate data acquisition and analysis in fields like telecommunications, control systems, and biomedical engineering, all of which are central to ENIB’s engineering programs. The ability to identify the specific aliased frequency demonstrates a deep grasp of the underlying principles of digital signal processing.

The core of this question lies in understanding the principles of signal processing, specifically the Nyquist-Shannon sampling theorem and its implications for digital representation of analog signals. The theorem states that to perfectly reconstruct an analog signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency component (\(f_{max}\)) present in the signal. This minimum sampling rate is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In this scenario, the analog signal has a maximum frequency component of 15 kHz. Therefore, according to the Nyquist-Shannon sampling theorem, the minimum sampling frequency required to avoid aliasing and ensure perfect reconstruction is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the *highest* frequency component that can be accurately represented by a sampling rate of 25 kHz. If the sampling frequency is less than the Nyquist rate, aliasing occurs, where higher frequencies in the original signal masquerade as lower frequencies in the sampled signal. The maximum frequency that can be unambiguously represented without aliasing is half the sampling frequency, which is the Nyquist frequency. So, with a sampling frequency \(f_s = 25 \text{ kHz}\), the highest frequency component that can be accurately represented is \(f_{max\_representable} = \frac{f_s}{2} = \frac{25 \text{ kHz}}{2} = 12.5 \text{ kHz}\). Any frequency component in the original analog signal above 12.5 kHz would be aliased to a lower frequency, leading to distortion and loss of information. This concept is fundamental in digital signal processing, a key area of study at institutions like ENIB, which emphasizes the practical application of theoretical principles in fields like telecommunications and embedded systems. Understanding the limits of digital representation is crucial for designing effective signal acquisition systems and avoiding data corruption.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the aliasing phenomenon and its mitigation through anti-aliasing filters. Aliasing occurs when a signal is sampled at a rate lower than twice its highest frequency component (Nyquist rate). If a signal contains frequencies above half the sampling rate, these higher frequencies will be misrepresented as lower frequencies in the sampled data, leading to distortion. Consider a signal with a maximum frequency component \(f_{max}\). To avoid aliasing, the sampling frequency \(f_s\) must satisfy \(f_s > 2 \cdot f_{max}\). If the sampling frequency is \(f_s = 8000\) Hz, then the maximum frequency that can be accurately represented without aliasing is \(f_{Nyquist} = f_s / 2 = 8000 \text{ Hz} / 2 = 4000\) Hz. The scenario describes a signal containing frequencies up to \(6000\) Hz. If this signal is sampled at \(8000\) Hz, the frequencies between \(4000\) Hz and \(6000\) Hz will alias. Specifically, a frequency \(f\) in this range will appear as \(|f – k \cdot f_s|\) for some integer \(k\), such that the aliased frequency is within the \(0\) to \(f_s/2\) range. For a frequency of \(6000\) Hz, with \(f_s = 8000\) Hz, the aliased frequency is \(|6000 – 1 \cdot 8000| = |-2000| = 2000\) Hz. To prevent this aliasing, an anti-aliasing filter is used before sampling. This filter is a low-pass filter designed to attenuate or remove frequencies above the Nyquist frequency (\(f_s/2\)) present in the analog signal before it is digitized. Therefore, to accurately sample a signal with a maximum frequency of \(6000\) Hz using a sampling rate of \(8000\) Hz, the anti-aliasing filter must effectively remove or significantly attenuate all frequencies above \(4000\) Hz. This ensures that only frequencies within the range of \(0\) to \(4000\) Hz are present when the signal is sampled, thus preventing aliasing. The crucial characteristic of such a filter is its cutoff frequency, which should be set at or below the Nyquist frequency.

The question probes the understanding of fundamental principles in digital signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications for aliasing. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a signal from its samples, the sampling frequency \(f_s\) must be at least twice the highest frequency component \(f_{max}\) present in the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_{Nyquist} = 2f_{max}\). In this scenario, a continuous-time signal with a maximum frequency component of \(15 \text{ kHz}\) is being sampled. To avoid aliasing, which is the distortion that occurs when the sampling frequency is too low, the sampling frequency must be greater than or equal to twice the maximum frequency. Therefore, the minimum required sampling frequency is \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). The question asks about the consequence of sampling at \(25 \text{ kHz}\). Since \(25 \text{ kHz} < 30 \text{ kHz}\), aliasing will occur. Aliasing causes higher frequency components in the original signal to be misrepresented as lower frequencies in the sampled signal. Specifically, a frequency \(f\) in the original signal will appear as \(|f – k f_s|\) for some integer \(k\), where \(f_s\) is the sampling frequency. For frequencies above \(f_s/2\), they will be folded back into the range \(0\) to \(f_s/2\). In this case, the \(15 \text{ kHz}\) component, which is greater than \(f_s/2 = 25 \text{ kHz} / 2 = 12.5 \text{ kHz}\), will be aliased. The aliased frequency will be \(|15 \text{ kHz} – 1 \times 25 \text{ kHz}| = |-10 \text{ kHz}| = 10 \text{ kHz}\). This means that the original \(15 \text{ kHz}\) signal component will be indistinguishable from a \(10 \text{ kHz}\) signal after sampling at \(25 \text{ kHz}\). This phenomenon is a critical consideration in the design of analog-to-digital converters for various engineering applications, including those studied at the National Engineering School of Brest Brittany ENIB, where accurate signal representation is paramount. Understanding and mitigating aliasing is a core competency for engineers working with digital systems.

The scenario describes a system where a signal is transmitted and received, with noise introduced during transmission. The core concept being tested is the ability to discern the original signal from the corrupted version, a fundamental challenge in signal processing and communications engineering, areas of significant focus at the National Engineering School of Brest Brittany ENIB. The question probes understanding of how signal-to-noise ratio (SNR) impacts the clarity and recoverability of information. A higher SNR indicates that the signal power is significantly greater than the noise power, making the original signal easier to distinguish. Conversely, a low SNR means the noise is comparable to or greater than the signal, leading to distortion and potential loss of information. In this context, the ability to accurately reconstruct the original waveform from the noisy observation is directly proportional to the SNR. Therefore, a system designed to maximize the fidelity of signal reconstruction under noisy conditions would prioritize maximizing the SNR. This involves techniques like filtering, amplification of the signal relative to the noise, and robust encoding/decoding schemes. The question requires an understanding that improving the SNR is the most direct and effective way to enhance the quality of the reconstructed signal, as it directly addresses the fundamental problem of signal degradation by noise. The other options, while potentially related to signal processing, do not directly address the core issue of signal clarity in the presence of additive noise as effectively as improving the SNR. For instance, increasing the sampling rate is beneficial for capturing high-frequency components but doesn’t inherently reduce noise. Reducing the bandwidth might filter out some noise but could also attenuate desired signal frequencies. Implementing error correction codes is crucial for detecting and correcting errors introduced by noise, but it’s a post-detection or transmission strategy rather than a direct improvement of the signal’s quality at the receiver’s input.

The scenario describes a system where a digital signal is processed. The core of the question lies in understanding the implications of aliasing in digital signal processing, a concept fundamental to courses at the National Engineering School of Brest Brittany ENIB. Aliasing occurs when the sampling rate is not sufficiently high relative to the highest frequency component in the analog signal, leading to misrepresentation of the original signal’s frequencies. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct an analog signal from its samples, the sampling frequency (\(f_s\)) must be at least twice the highest frequency (\(f_{max}\)) present in the signal. Mathematically, this is expressed as \(f_s \ge 2f_{max}\). In the given scenario, the analog signal contains frequencies up to 15 kHz. The sampling frequency is 20 kHz. According to the Nyquist-Shannon theorem, the minimum required sampling frequency to avoid aliasing for a signal with a maximum frequency of 15 kHz would be \(2 \times 15 \text{ kHz} = 30 \text{ kHz}\). Since the actual sampling frequency (20 kHz) is less than the required minimum (30 kHz), aliasing will occur. When aliasing happens, frequencies above \(f_s/2\) (the Nyquist frequency) are folded back into the frequency range below \(f_s/2\). The Nyquist frequency in this case is \(20 \text{ kHz} / 2 = 10 \text{ kHz}\). Frequencies in the analog signal that are greater than 10 kHz will appear as lower frequencies in the sampled digital signal. Specifically, a frequency \(f\) greater than \(f_s/2\) will appear as \(|f – n \cdot f_s|\) for some integer \(n\), such that the resulting frequency is within the range \([0, f_s/2]\). For the 15 kHz component, which is above the Nyquist frequency of 10 kHz, it will alias. The closest frequency below 10 kHz that it will appear as is \(|15 \text{ kHz} – 1 \cdot 20 \text{ kHz}| = |-5 \text{ kHz}| = 5 \text{ kHz}\). Therefore, the 15 kHz component will be incorrectly represented as a 5 kHz signal in the digital domain. This distortion means the reconstructed analog signal will not accurately reflect the original signal’s spectral content, leading to an erroneous output. This understanding of sampling and aliasing is crucial for students at ENIB, particularly in fields like telecommunications, signal processing, and embedded systems design, where accurate digital representation of analog phenomena is paramount.

The core principle being tested here is the understanding of signal-to-noise ratio (SNR) in the context of data acquisition and processing, a fundamental concept in many engineering disciplines taught at ENIB, particularly in areas like signal processing, telecommunications, and instrumentation. While no direct calculation is required, the scenario implicitly involves the trade-offs in data quality. The scenario describes a sensor array designed for environmental monitoring. The goal is to detect subtle changes in atmospheric composition. The sensor’s inherent thermal noise contributes to the “noise” component of the SNR. The signal strength is directly related to the concentration of the target atmospheric component. To improve the ability to detect low concentrations (weak signals), one must increase the signal-to-noise ratio. This can be achieved by either increasing the signal strength or decreasing the noise. Consider the options: 1. **Increasing the sampling rate:** While a higher sampling rate can capture more data points, it doesn’t inherently reduce the noise floor of the sensor itself. If anything, it might increase the amount of noisy data collected per unit time. It doesn’t directly address the fundamental noise level of the sensor or the signal strength. 2. **Averaging multiple readings:** This is a standard technique to reduce the impact of random noise. If the noise is random and uncorrelated across multiple measurements, averaging \(N\) independent samples will reduce the noise variance by a factor of \(N\), and thus the standard deviation (and typically the noise power) by a factor of \(\sqrt{N}\). This effectively increases the SNR by \(\sqrt{N}\) without altering the true signal. This directly enhances the detection of weak signals. 3. **Using a lower-resolution analog-to-digital converter (ADC):** A lower-resolution ADC quantifies the analog signal with fewer bits, leading to coarser quantization steps. This introduces quantization error, which is a form of noise. Therefore, using a lower-resolution ADC would *decrease* the SNR, making it harder to detect weak signals. 4. **Implementing a high-pass filter:** A high-pass filter would attenuate frequencies below a certain cutoff. If the signal of interest (the atmospheric component’s concentration change) is at low frequencies, a high-pass filter would remove or weaken the signal itself, thus decreasing the SNR and hindering detection. Therefore, averaging multiple readings is the most effective strategy among the given options to improve the SNR and thus enhance the detection of subtle atmospheric changes. This aligns with ENIB’s emphasis on robust data acquisition and signal processing techniques for real-world applications.

The question probes the understanding of a fundamental concept in digital signal processing, specifically the Nyquist-Shannon sampling theorem and its implications for aliasing. The scenario describes a signal with a maximum frequency component of \(f_{max} = 15 \, \text{kHz}\). According to the Nyquist-Shannon sampling theorem, to perfectly reconstruct a signal, the sampling frequency \(f_s\) must be at least twice the maximum frequency component of the signal, i.e., \(f_s \ge 2f_{max}\). This minimum sampling frequency is known as the Nyquist rate. In this case, the Nyquist rate is \(2 \times 15 \, \text{kHz} = 30 \, \text{kHz}\). If the signal is sampled at a frequency lower than the Nyquist rate, aliasing will occur. Aliasing is the phenomenon where high-frequency components in the original signal are misrepresented as lower frequencies in the sampled signal, leading to distortion and loss of information. The question asks about the consequence of sampling at \(25 \, \text{kHz}\). Since \(25 \, \text{kHz} < 30 \, \text{kHz}\), aliasing will indeed occur. The specific frequency that a higher frequency component will alias to depends on the sampling frequency. A frequency \(f\) greater than \(f_s/2\) will alias to \(|f - k \cdot f_s|\), where \(k\) is an integer chosen such that the resulting frequency is within the range \([0, f_s/2]\). Consider a frequency component at \(15 \, \text{kHz}\). The folding frequency is \(f_s/2 = 25 \, \text{kHz} / 2 = 12.5 \, \text{kHz}\). Since \(15 \, \text{kHz} > 12.5 \, \text{kHz}\), this component will alias. The aliased frequency will be \(|15 \, \text{kHz} – 1 \cdot 25 \, \text{kHz}| = |-10 \, \text{kHz}| = 10 \, \text{kHz}\). This means the original \(15 \, \text{kHz}\) component will appear as a \(10 \, \text{kHz}\) component in the sampled signal. This is a direct violation of the sampling theorem and demonstrates the loss of fidelity. The core principle tested here is the direct application of the Nyquist criterion and the understanding of what happens when it is violated, a crucial concept for students at the National Engineering School of Brest Brittany ENIB, particularly in fields involving signal processing and telecommunications. The ability to identify and predict aliasing is fundamental for designing effective digital systems and avoiding data corruption.

The question probes the understanding of the fundamental principles of signal processing, specifically concerning the Nyquist-Shannon sampling theorem and its implications in digital signal reconstruction. The scenario describes a continuous-time signal \(x(t)\) with a maximum frequency component of \(f_{max} = 150\) Hz. The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a continuous-time signal from its samples, the sampling frequency \(f_s\) must be at least twice the maximum frequency component of the signal. This minimum sampling frequency is known as the Nyquist rate, \(f_{Nyquist} = 2 \times f_{max}\). In this case, \(f_{max} = 150\) Hz. Therefore, the Nyquist rate is \(f_{Nyquist} = 2 \times 150\) Hz \(=\) 300 Hz. The question asks about the minimum sampling frequency required to avoid aliasing and ensure faithful reconstruction. According to the theorem, this minimum frequency is the Nyquist rate. Thus, the required sampling frequency \(f_s\) must satisfy \(f_s \ge f_{Nyquist}\). The smallest such \(f_s\) is exactly \(f_{Nyquist}\). Therefore, the minimum sampling frequency required is 300 Hz. This concept is crucial in digital signal processing, a core area within many engineering disciplines taught at the National Engineering School of Brest Brittany (ENIB), particularly in fields like telecommunications, control systems, and embedded systems. Understanding aliasing and the conditions for perfect reconstruction is essential for designing effective digital filters, analog-to-digital converters (ADCs), and digital-to-analog converters (DACs), ensuring that the digital representation accurately captures the information present in the original analog signal. Failure to adhere to the Nyquist criterion can lead to distortion and loss of vital signal information, rendering the reconstructed signal unusable for its intended purpose. The ENIB’s emphasis on practical applications and theoretical rigor means that candidates are expected to grasp these foundational concepts thoroughly.