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Quiz: Convolution and Correlation

Test your understanding of convolution and correlation operations in signal processing.

1. What is the continuous-time convolution integral formula?

  1. \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t+\tau) d\tau\)
  2. \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau\)
  3. \(y(t) = \int_{-\infty}^{\infty} x(t)h(\tau) d\tau\)
  4. \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(\tau) d\tau\)
Show Answer

The correct answer is B. The convolution integral is defined as \(y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau\), which can also be written as \(y(t) = x(t) * h(t)\). This operation computes the overlap between one signal and a time-reversed, shifted version of the other signal. For LTI systems, \(x(t)\) is the input signal, \(h(t)\) is the impulse response, and \(y(t)\) is the output.

Concept Tested: Convolution

See: Convolution

2. What is the discrete convolution formula?

  1. \(y[n] = \sum_{k=-\infty}^{\infty} x[k]h[k-n]\)
  2. \(y[n] = \sum_{k=-\infty}^{\infty} x[n]h[k]\)
  3. \(y[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\)
  4. \(y[n] = \prod_{k=-\infty}^{\infty} x[k]h[n-k]\)
Show Answer

The correct answer is C. Discrete convolution is defined as \(y[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k]\), which is the discrete-time equivalent of the continuous convolution integral. This formula expresses the output of a discrete-time LTI system as a weighted sum of present and past input samples, and appears frequently in FIR filter implementations.

Concept Tested: Discrete Convolution

See: Discrete Convolution

3. What is the key difference between circular convolution and linear convolution?

  1. Circular convolution uses multiplication instead of summation
  2. Circular convolution treats signals as periodic with period \(N\), wrapping indices modulo \(N\)
  3. Circular convolution is faster but less accurate than linear convolution
  4. Circular convolution can only be used for real-valued signals
Show Answer

The correct answer is B. Circular convolution treats signals as periodic with period \(N\), so samples "wrap around" from the end back to the beginning using modulo arithmetic: \(y[n] = \sum_{k=0}^{N-1} x[k]h[(n-k) \bmod N]\). This operation arises naturally when computing convolution using the DFT, which implicitly assumes periodic signals. To obtain linear convolution results using the DFT, signals must be zero-padded to length at least \(N_1 + N_2 - 1\).

Concept Tested: Circular Convolution

See: Circular Convolution

4. According to the convolution theorem, what does convolution in the time domain correspond to in the frequency domain?

  1. Addition: \(\mathcal{F}\{x(t) * h(t)\} = X(f) + H(f)\)
  2. Convolution: \(\mathcal{F}\{x(t) * h(t)\} = X(f) * H(f)\)
  3. Division: \(\mathcal{F}\{x(t) * h(t)\} = X(f) / H(f)\)
  4. Multiplication: \(\mathcal{F}\{x(t) * h(t)\} = X(f) \cdot H(f)\)
Show Answer

The correct answer is D. The convolution theorem states that convolution in the time domain corresponds to multiplication in the frequency domain: \(\mathcal{F}\{x(t) * h(t)\} = X(f) \cdot H(f)\). This fundamental relationship enables efficient filtering by multiplying in the frequency domain and transforming back, often more efficient than direct time-domain convolution for large filters.

Concept Tested: Convolution Theorem

See: Convolution Theorem

5. How does correlation differ from convolution in its basic operation?

  1. Correlation uses addition while convolution uses multiplication
  2. Correlation slides \(y(t)\) without time reversal; convolution uses \(h(t-\tau)\) reversed
  3. Correlation is only defined for discrete-time signals
  4. Correlation produces complex-valued outputs while convolution produces real outputs
Show Answer

The correct answer is B. The key difference is that correlation slides \(y(t)\) without time reversal (using \(y(t-τ)\)), whereas convolution uses the time-reversed function \(h(t-τ)\). The relationship between them is \(R_{xy}(τ) = x(τ) * y(-τ)\), showing that correlation can be computed as convolution with one signal time-reversed.

Concept Tested: Correlation, Convolution

See: Correlation

6. What is a key property of the autocorrelation function \(R_{xx}(τ)\)?

  1. It is always zero at zero lag
  2. It achieves its minimum value at zero lag
  3. It is symmetric about zero lag: \(R_{xx}(τ) = R_{xx}(-τ)\)
  4. It is only defined for periodic signals
Show Answer

The correct answer is C. The autocorrelation function is always symmetric about zero lag: \(R_{xx}(τ) = R_{xx}(-τ)\). Additionally, it achieves its maximum value at zero lag: \(R_{xx}(0) \geq |R_{xx}(τ)|\) for all \(τ\). Autocorrelation reveals the internal structure of a signal, particularly periodic components and self-similarity across different time scales.

Concept Tested: Autocorrelation

See: Autocorrelation

7. What range of values can the normalized correlation coefficient \(\rho_{xy}\) take?

  1. \([0, \infty)\)
  2. \([-1, 1]\)
  3. \([0, 1]\)
  4. \((-\infty, \infty)\)
Show Answer

The correct answer is B. The correlation coefficient is normalized to the range \([-1, 1]\). A value of \(\rho_{xy} = 1\) indicates perfect positive correlation (signals are identical up to scaling), \(\rho_{xy} = -1\) indicates perfect negative correlation (one signal is the negative of the other), and \(\rho_{xy} = 0\) indicates no linear relationship (signals are orthogonal or uncorrelated).

Concept Tested: Correlation Coefficient

See: Correlation Coefficient

8. What is the impulse response of a matched filter designed to detect signal \(s(t)\) in noise?

  1. \(h(t) = s(t)\)
  2. \(h(t) = s^*(T - t)\) (time-reversed complex conjugate)
  3. \(h(t) = s(T + t)\)
  4. \(h(t) = |s(t)|\)
Show Answer

The correct answer is B. The matched filter has impulse response \(h(t) = s^*(T - t)\), which is the time-reversed complex conjugate of the signal to be detected, where \(T\) is the time at which SNR is maximized. The matched filter is optimal for detecting a known signal in additive white noise, maximizing the signal-to-noise ratio. It essentially performs correlation between the received signal and the known template.

Concept Tested: Matched Filter

See: Matched Filter

9. If you convolve two finite-length discrete sequences of lengths \(N_1 = 10\) and \(N_2 = 15\), what is the length of the linear convolution output?

  1. \(10\) samples
  2. \(15\) samples
  3. \(24\) samples
  4. \(25\) samples
Show Answer

The correct answer is C. The length of the linear convolution output is \(N_1 + N_2 - 1 = 10 + 15 - 1 = 24\) samples. This is important when using the DFT for fast convolution: to avoid circular convolution artifacts, signals must be zero-padded to at least this length before applying the DFT-multiply-IDFT procedure.

Concept Tested: Discrete Convolution, Linear Convolution

See: Discrete Convolution, Circular Convolution

10. What optimization criterion does the Wiener filter minimize?

  1. Maximum absolute error
  2. Total signal energy
  3. Mean square error between filter output and desired signal
  4. Maximum signal-to-noise ratio at a specific time instant
Show Answer

The correct answer is C. The Wiener filter is the optimal linear filter for estimating a desired signal from a noisy observation by minimizing the mean square error between the filter output and the desired signal. The frequency response is \(H(f) = S_{ds}(f)/S_{xx}(f)\), balancing noise suppression against signal distortion. Note that option D describes the matched filter, not the Wiener filter.

Concept Tested: Wiener Filter

See: Wiener Filter