Correlation and cross-correlation in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of cross-correlation between two signals

What is the output of the following Python code that computes the cross-correlation of two signals using numpy?

Signal Processing

import numpy as np
x = np.array([1, 2, 3])
y = np.array([0, 1, 0.5])
result = np.correlate(x, y, mode='full')
print(result.tolist())

A [1.0, 2.0, 3.0]

B [0.0, 3.0, 3.5, 2.0, 0.5]

C [0.0, 1.0, 2.0, 3.0, 4.0]

D [0.5, 2.0, 3.5, 3.0, 0.0]

💡 Hint

Recall that numpy.correlate with mode='full' returns the cross-correlation at all lags.

The cross-correlation of x and y with mode='full' computes the sum of products at each lag, resulting in [0.5, 2.0, 3.5, 3.0, 0.0].

❓ data_output

intermediate

Length of correlation output array

Given two signals of lengths 7 and 5, what is the length of their cross-correlation output using full mode?

A 11

B 12

C 7

D 13

💡 Hint

The length of full cross-correlation output is length_x + length_y - 1.

For signals of length 7 and 5, the full cross-correlation length is 7 + 5 - 1 = 11.

🧠 Conceptual

advanced

Understanding correlation symmetry

Which statement correctly describes the symmetry property of cross-correlation between two real signals x and y?

A Cross-correlation of x and y equals the convolution of x and y.

B Cross-correlation of x and y is always symmetric around zero lag.

C Cross-correlation of x and y equals cross-correlation of y and x reversed in time.

D Cross-correlation of x and y is always zero at zero lag.

💡 Hint

Think about how swapping signals affects the lag axis.

Cross-correlation of x and y equals the cross-correlation of y and x flipped in time (reversed). This is a key symmetry property.

🔧 Debug

advanced

Identify the error in cross-correlation code

What error will the following code produce when computing cross-correlation of two lists?

Signal Processing

import numpy as np
x = [1, 2, 3]
y = [0, 1, 0.5]
result = np.correlate(x, y, mode='full')
print(result)

A No error, prints cross-correlation array

B ValueError due to incompatible input lengths

C SyntaxError due to missing parentheses

D TypeError because inputs are lists, not arrays

💡 Hint

Check if numpy.correlate accepts lists as input.

numpy.correlate accepts lists and converts them internally to arrays, so no error occurs and it prints the cross-correlation.

🚀 Application

expert

Detecting time delay using cross-correlation

You have two signals: a clean signal and a delayed noisy version. Which method correctly estimates the delay between them using cross-correlation?

A Find the lag index where cross-correlation between signals is minimum.

B Find the lag index where cross-correlation between signals is maximum.

C Compute the sum of cross-correlation values and divide by signal length.

D Use the zero lag cross-correlation value as the delay estimate.

💡 Hint

The delay corresponds to the lag with highest similarity.

The time delay is estimated by finding the lag where the cross-correlation reaches its maximum, indicating best alignment.

Practice

1. What does the cross-correlation between two signals primarily measure?

easy

A. The similarity between the signals at different time shifts

B. The sum of the signals' amplitudes

C. The difference in frequency between the signals

D. The product of the signals' maximum values

Solution

1. Step 1: Understand cross-correlation concept

   Cross-correlation compares two signals by sliding one over the other to find similarity at different shifts.

2. Step 2: Identify what cross-correlation measures

   It measures how well the signals match at each time delay, not just their sums or frequency differences.

3. Final Answer:

   The similarity between the signals at different time shifts -> Option A

4. Quick Check:

   Cross-correlation = similarity at shifts [OK]

Hint: Cross-correlation slides signals to find best match [OK]

Common Mistakes:

• Confusing correlation with frequency difference
• Thinking it sums amplitudes only
• Assuming it multiplies max values

2. Which Python function from NumPy computes the cross-correlation of two 1D signals correctly?

easy

A. numpy.convolve(signal1, signal2)

B. numpy.correlate(signal1, signal2, mode='full')

C. numpy.cross(signal1, signal2)

D. numpy.corrcoef(signal1, signal2)

Solution

1. Step 1: Identify NumPy functions for correlation

   numpy.correlate computes cross-correlation; numpy.cross is for vector cross product; numpy.convolve does convolution; numpy.corrcoef computes correlation coefficient matrix.

2. Step 2: Confirm correct function and parameters

   numpy.correlate with mode='full' returns cross-correlation at all shifts.

3. Final Answer:

   numpy.correlate(signal1, signal2, mode='full') -> Option B

4. Quick Check:

   Use numpy.correlate for cross-correlation [OK]

Hint: Use numpy.correlate with mode='full' for cross-correlation [OK]

Common Mistakes:

• Using numpy.cross which is vector product
• Confusing convolution with correlation
• Using numpy.corrcoef which gives correlation matrix

3. Given two signals x = [1, 2, 3] and y = [0, 1, 0.5], what is the cross-correlation output using numpy.correlate(x, y, mode='full')?

medium

A. [0.5, 2.5, 4, 3, 0]

B. [0, 1, 2, 3, 4]

C. [0.5, 2, 3.5, 3, 0]

D. [1, 2, 3, 0, 0]

Solution

1. Step 1: Calculate cross-correlation values manually

   Cross-correlation at lag -2: x[0]*y[2]=1*0.5=0.5; lag -1: x[0]*y[1]+x[1]*y[2]=1*1+2*0.5=1+1=2; lag 0: x[0]*y[0]+x[1]*y[1]+x[2]*y[2]=1*0+2*1+3*0.5=0+2+1.5=3.5; lag 1: x[1]*y[0]+x[2]*y[1]=2*0+3*1=3; lag 2: x[2]*y[0]=3*0=0

2. Step 2: Arrange results in full mode order

   Full mode output is [lag -2, -1, 0, 1, 2] = [0.5, 2, 3.5, 3, 0]

3. Final Answer:

   [0.5, 2, 3.5, 3, 0] -> Option C

4. Quick Check:

   Cross-correlation values match calculated array [OK]

Hint: Calculate lag by lag multiplying overlapping elements [OK]

Common Mistakes:

• Mixing convolution with correlation
• Incorrect lag order in output
• Forgetting zero padding in full mode

4. You run numpy.correlate(signal1, signal2) but get a shorter output than expected. What is the likely cause?

medium

A. The function requires 2D arrays, not 1D

B. signal1 and signal2 must be the same length

C. You need to normalize signals before correlation

D. The default mode is 'valid', which returns only overlapping parts

Solution

1. Step 1: Check default mode of numpy.correlate

   By default, numpy.correlate uses mode='valid', which returns only positions where signals fully overlap, resulting in shorter output.

2. Step 2: Understand effect of mode on output length

   To get full cross-correlation including partial overlaps, mode='full' must be specified.

3. Final Answer:

   The default mode is 'valid', which returns only overlapping parts -> Option D

4. Quick Check:

   Default mode='valid' shortens output [OK]

Hint: Specify mode='full' for complete cross-correlation output [OK]

Common Mistakes:

• Assuming signals must be same length
• Thinking normalization affects output length
• Believing function needs 2D arrays

5. You have two noisy signals recorded with a delay. How can cross-correlation help you find the delay between them?

hard

A. By finding the lag where cross-correlation is maximum, indicating best alignment

B. By subtracting the signals and measuring the difference

C. By computing the Fourier transform and comparing magnitudes

D. By averaging the signals and checking the mean value

Solution

1. Step 1: Understand cross-correlation use for delay detection

   Cross-correlation slides one signal over the other and measures similarity at each lag.

2. Step 2: Identify lag with maximum cross-correlation

   The lag where cross-correlation peaks shows the delay that best aligns the two signals despite noise.

3. Final Answer:

   By finding the lag where cross-correlation is maximum, indicating best alignment -> Option A

4. Quick Check:

   Max cross-correlation lag = signal delay [OK]

Hint: Max cross-correlation lag reveals signal delay [OK]

Common Mistakes:

• Using subtraction which ignores alignment
• Confusing frequency domain methods here
• Averaging signals loses timing info