Q. Sampling's importance in signal processing
A. To convert an analog signal to a digital one, sampling is required.
Q. Describe the Nyquist criterion.
A. When a message signal's highest frequency component is Fm, the sampling rate must be at least 2*Fm. If not, aliasing occurs and we cannot recover the original signal.
Q. Why do we need "signal quantization"?
A. Quantization maps a continuous signal to a discrete set of levels. Without it, the signal would have infinite possible values, making digital processing and storage impossible.
Q. Modulation definition
Q. Why is modulation necessary?
A. It reduces antenna size, allows multiplexing, and increases the range of transmission by shifting the signal to higher frequencies.
Q. What are heterodyne or super-heterodyne receivers?
A. These receivers mix the incoming high-frequency signal with a local oscillator frequency to convert it to a fixed, lower Intermediate Frequency (IF) for easier processing.
Q. What is a communication channel's capacity?
where BW = bandwidth in Hz.
Q. A constellation diagram is what?
Q. Draw the BPSK constellation diagram (binary phase shift keying).
A. It consists of two points on the horizontal (I) axis at 0 and 180 degrees.
Q. Draw the 16 QAM constellation diagram (quadrature amplitude modulation).
Q. What do baud rate and bit rate mean?
A. Baud rate is the number of symbols sent per second. Bit rate is the number of bits sent per second (Bit Rate = Baud Rate * bits per symbol).
Q. Coherence bandwidth: what is it?
Q. Which QAM format—32 QAM, 64 QAM, 128 QAM, and 256 QAM—has the most bits per symbol?
64 QAM -----> 6 bits per symbol
256 QAM -----> 8 bits per symbol
Q. Which modulation scheme—4 QAM or BPSK—will you choose in a noisy channel?
A. BPSK is preferred in very noisy channels because the distance between constellation points is larger, providing better noise immunity.
Q. Assume you are using two different modulation techniques. Both BPSK and QPSK modulation are used in the first. Please tell us what the SNR for QPSK should be when the SNR for BPSK is 5 dB to have the same BER (bit error rate) for both modulation techniques.
A. For the same BER, QPSK requires approximately 3 dB more signal power (SNR) than BPSK because QPSK transmits two bits per symbol.
Q. OFDM modulation: what is it? What are the benefits of OFDM modulation?
A. OFDM splits a high-rate data stream into multiple slow-rate subcarriers. Benefits include high spectral efficiency and robustness against multi-path fading.
Q. Alamouti scheme: what is it?
A. Alamouti scheme is a space-time block coding technique that sends two orthogonal symbols from two transmitters at two different time slots. It provides transmit diversity.
Stochastic Processes & Signal Theory
Q. What defines a Wide-Sense Stationary (WSS) process? A. A process is WSS if its mean is constant over time and its autocorrelation function depends only on the time lag ($\tau$), not on absolute time. Note: Unlike strict stationarity, WSS only requires the first and second moments (mean and variance) to be time-invariant. Q. In LTI systems, what is the significance of the convolution operation? A. Convolution in the time domain represents the relationship between the input signal, the system's impulse response, and the resulting output. Mathematically, convolution in the time domain is equivalent to point-wise multiplication in the frequency domain. Q. Explain the difference between Autoregressive (AR) and Moving Average (MA) processes. A. An Autoregressive (AR) process models the current value of a signal as a linear combination of its past values plus noise. A Moving Average (MA) process models the current value based on past white noise (error) terms. Q. What is Ergodicity in the context of a WSS process? A. An ergodic process is one where statistical properties (like mean and variance) can be determined from a single, sufficiently long realization of the process. In short, "time averages equal ensemble averages."System Stability & ARMA Modeling
Q. Under what conditions is an ARMA model both causal and invertible? A. For causality, all poles of the AR polynomial must lie inside the unit circle in the Z-domain. For invertibility, all zeros of the MA polynomial must also lie inside the unit circle. Q. What happens to an LTI system if its poles lie outside the unit circle? A. The system becomes unstable. Any input will cause the output to grow without bound (diverge), as the impulse response does not decay over time. Q. What is "White Noise" in time-series analysis? A. White noise is a random process with a zero mean, constant variance, and zero autocorrelation for any non-zero lag. Its Power Spectral Density (PSD) is flat across all frequencies, similar to white light. Q. Explain the effect of "Pole-Zero Cancellation" in an ARMA(1,1) model. A. If a pole and a zero are located at the same point in the Z-domain, they cancel each other out, simplifying the system's complexity and effectively reducing the order of the model. Q. How does the 'a' parameter in an AR(1) process affect its Power Spectral Density (PSD)? A. As the absolute value of the coefficient $|a|$ approaches 1, the process becomes more correlated (longer memory). The PSD exhibits a sharp peak at low frequencies (low-pass characteristic) and the process approaches non-stationarity (random walk). Q. Why is the ARMA model preferred for stationary time series? A. ARMA is parsimonious; it can model complex dependencies with fewer parameters by combining both autoregressive (memory) and moving average (noise smoothing) components, whereas a pure AR or MA model might require many more terms to achieve the same accuracy.Fourier Transform & Signal Properties
Q. What is the Fourier Transform of the Dirac delta function $\delta(t)$? A. The Fourier Transform of $\delta(t)$ is a constant function (1) across all frequencies. This implies that an impulse contains all frequencies with equal weight. Q. What is the significance of the Duality property in Fourier Transforms? A. Duality states that if $x(t) \leftrightarrow X(f)$, then $X(t) \leftrightarrow x(-f)$. This symmetry allows us to find the transform of a signal in one domain if we already know its inverse in the other. Q. What time-domain signal corresponds to a Sinc function in the frequency domain? A. A rectangular pulse. Due to the inverse relationship, a rectangular pulse in time results in a Sinc function in frequency, and a band-limited (rectangular) spectrum results in a Sinc pulse in time. Q. Can a signal be both time-limited and band-limited? A. No. A fundamental principle of Fourier analysis is that a signal cannot be perfectly limited in both time and frequency. If a signal is finite in duration (time-limited), its Fourier Transform will have infinite support (infinite bandwidth). Q. What is the mathematical effect of multiplying a signal by a cosine wave in the time domain? A. This is Modulation. Multiplying $x(t)$ by $\cos(2\pi f_c t)$ shifts the signal's spectrum to be centered at $\pm f_c$. This is the basis of up-conversion in transmitters. Q. If a signal is compressed in time ($x(at)$ where $a > 1$), what happens to its spectrum? A. The spectrum expands in frequency and its amplitude decreases. Mathematically: $x(at) \leftrightarrow \frac{1}{|a|}X(\frac{f}{a})$. Compression in time is expansion in frequency. Q. What is the Fourier Transform of a signal that is both real and even? A. The Fourier Transform of a real and even signal is also real and even. If the signal is real and odd, its transform is purely imaginary and odd. Q. What causes "Spectral Leakage" when using a rectangular window? A. Truncating a signal abruptly with a rectangular window is equivalent to multiplying by a pulse. In the frequency domain, this is a convolution with a Sinc function, which introduces "sidelobes" or ripples that leak energy into adjacent frequencies. Q. Explain the Convolution Theorem. A. The Convolution Theorem states that convolution in the time domain corresponds to point-wise multiplication in the frequency domain. This makes it much easier to analyze the output of a system (Filter) by simply multiplying the input spectrum by the filter's transfer function.Sampling & Discrete Systems
Q. What happens to the Fourier Transform of a signal when it is sampled at rate $f_s$? A. Sampling in the time domain makes the spectrum periodic in the frequency domain, with a period equal to the sampling frequency $f_s$. If $f_s$ is less than twice the highest frequency (Nyquist), these periodic replicas overlap, causing aliasing. Q. What is the Fourier Transform of the Unit Step function $u(t)$? A. In the angular frequency domain, it is expressed as $U(\omega) = \pi\delta(\omega) + \frac{1}{j\omega}$. The delta function represents the DC component (average value of 0.5), and the $1/j\omega$ represents the transient change. Q. What is the Fourier Transform of a triangular pulse? A. The Fourier Transform of a triangular pulse is a Sinc squared function $[\text{sinc}(f)]^2$. This is because a triangular pulse is the result of convolving two rectangular pulses.[1] Important wireless communication terms