Correlation and cross-correlation in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute the correlation of a signal with itself using numpy.

Signal Processing

import numpy as np
signal = np.array([1, 2, 3, 4, 5])
correlation = np.correlate(signal, signal, [1])
print(correlation)

Drag options to blanks, or click blank then click option'

A"same"

B"full"

C"valid"

D"centered"

Attempts:

3 left

💡 Hint

Common Mistakes

Using mode "same" or "valid" which returns shorter correlation arrays.

Passing an invalid mode string causing an error.

The np.correlate function with mode "full" returns the complete cross-correlation, including all lags.

2fill in blank

medium

Complete the code to compute the cross-correlation between two signals using numpy.

Signal Processing

import numpy as np
signal1 = np.array([1, 2, 3])
signal2 = np.array([0, 1, 0.5])
cross_corr = np.correlate(signal1, signal2, [1])
print(cross_corr)

Drag options to blanks, or click blank then click option'

A"valid"

B"centered"

C"same"

D"full"

Attempts:

3 left

💡 Hint

Common Mistakes

Using mode "valid" which returns only positions where signals fully overlap.

Using an invalid mode string causing an error.

Using mode "full" computes the cross-correlation at all lags between the two signals.

3fill in blank

hard

Fix the error in the code to compute normalized cross-correlation between two signals.

Signal Processing

import numpy as np
signal1 = np.array([1, 2, 3, 4])
signal2 = np.array([0, 1, 0, 1])
correlation = np.correlate(signal1 - np.mean(signal1), signal2 - np.mean(signal2), [1])
normalization = np.std(signal1) * np.std(signal2) * len(signal1)
normalized_corr = correlation / normalization
print(normalized_corr)

Drag options to blanks, or click blank then click option'

A"same"

B"full"

C"valid"

D"centered"

Attempts:

3 left

💡 Hint

Common Mistakes

Using mode "full" or "same" which includes partial overlaps.

Forgetting to subtract the mean before correlation.

The "valid" mode returns correlation only where the signals fully overlap, which is appropriate for normalized cross-correlation.

4fill in blank

hard

Fill both blanks to create a dictionary of cross-correlation values for lags from -2 to 2.

Signal Processing

import numpy as np
signal1 = np.array([1, 2, 3, 4, 5])
signal2 = np.array([5, 4, 3, 2, 1])
correlations = {lag: np.correlate(signal1, np.roll(signal2, lag), [1])[0] for lag in range(-2, 3) if lag [2] 0}
print(correlations)

Drag options to blanks, or click blank then click option'

A"valid"

B==

C!=

D>

Attempts:

3 left

💡 Hint

Common Mistakes

Using mode "full" which includes partial overlaps.

Using wrong condition like lag == 0 which includes only zero lag.

Using mode "valid" ensures correlation at full overlap. The condition lag != 0 excludes zero lag from the dictionary.

5fill in blank

hard

Fill all three blanks to compute normalized cross-correlation and store results for lags -1, 0, 1.

Signal Processing

import numpy as np
signal1 = np.array([2, 3, 4])
signal2 = np.array([1, 0, 1])
results = {}
for lag in range(-1, 2):
    rolled = np.roll(signal2, lag)
    corr = np.correlate(signal1 - np.mean(signal1), rolled - np.mean(rolled), [1])[0]
    norm = np.std(signal1) * np.std(rolled) * [2]
    results[[3]] = corr / norm
print(results)

Drag options to blanks, or click blank then click option'

A"valid"

Blen(signal1)

Clag

Dlen(signal2)

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong mode causing incorrect correlation length.

Normalizing by wrong length or forgetting to normalize.

Using wrong dictionary key causing overwriting results.

Use mode "valid" for full overlap correlation. Normalize by multiplying standard deviations and length of signal1. Store results keyed by the current lag.

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