Correlation and cross-correlation in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is correlation in signal processing?

Correlation measures how similar two signals are as one shifts over the other. It shows if patterns in one signal match patterns in another.

beginner

What does cross-correlation tell us between two signals?

Cross-correlation shows how much one signal matches another at different time shifts. It helps find delays or similarities between signals.

intermediate

How is auto-correlation different from cross-correlation?

Auto-correlation compares a signal with itself at different shifts to find repeating patterns. Cross-correlation compares two different signals.

intermediate

Why is normalization important in correlation?

Normalization scales correlation values between -1 and 1, making it easier to compare strength of similarity regardless of signal size.

beginner

What does a correlation value close to 0 mean?

A correlation near 0 means there is little or no linear similarity between the two signals at that shift.

What does cross-correlation help you find between two signals?

A The noise level in the signals

B The frequency of the signals

C The amplitude of the first signal

D The time delay where they best match

Cross-correlation finds the time shift where two signals are most similar, revealing delays.

Auto-correlation is used to detect what in a signal?

A Repeating patterns or periodicity

B Noise spikes

C Signal amplitude

D Cross-signal similarity

Auto-correlation compares a signal with itself to find repeating patterns or cycles.

If correlation between two signals is -1, what does it mean?

A They are perfectly negatively related

B They are identical

C They have no relation

D They are perfectly positively related

A correlation of -1 means the signals are perfectly opposite in pattern.

Why do we normalize correlation values?

A To increase signal amplitude

B To remove noise

C To compare similarity strength fairly

D To change signal frequency

Normalization scales values so we can fairly compare how strong the similarity is.

Which of these is NOT a use of cross-correlation?

A Finding time delay between signals

B Detecting repeating patterns in one signal

C Matching patterns between two signals

D Measuring similarity at different shifts

Detecting repeating patterns in one signal is auto-correlation, not cross-correlation.

Explain in your own words what cross-correlation is and how it can be used in signal processing.

Think about sliding one signal over another and measuring how well they match.

Describe the difference between auto-correlation and cross-correlation with examples.

Consider when you want to find patterns inside one signal versus comparing two signals.

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