Convolution in MATLAB - Step-by-Step Execution

Concept Flow - Convolution

Start: Two signals x and h

↓

Flip h to get h(-t)

↓

Shift h(-t) by t: h(t - tau)

↓

Multiply x(tau) * h(t - tau)

↓

Integrate over tau (sum in discrete)

↓

Output y(t) = convolution result

↓

End

Convolution combines two signals by flipping, shifting, multiplying, and summing to produce a new signal.

Execution Sample

MATLAB

x = [1 2 3];
h = [0 1 0.5];
y = conv(x, h);
disp(y);

This code convolves two vectors x and h, then displays the result.

Execution Table

Step	Index	x values	h values (flipped and shifted)	Multiplication	Sum at step	y output
1	1	1	0	1*0=0	0	0
2	2	1 2	1 0	1*1=1, 2*0=0	1	1
3	3	1 2 3	0.5 1 0	1*0.5=0.5, 2*1=2, 3*0=0	2.5	2.5
4	4	2 3	0.5 1	2*0.5=1, 3*1=3	4	4
5	5	3	0.5	3*0.5=1.5	1.5	1.5
6	End	-	-	-	-	Convolution complete, output length = length(x)+length(h)-1 = 5

💡 All overlapping positions processed; convolution output length is 5.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
x	[1 2 3]	[1 2 3]	[1 2 3]	[1 2 3]	[1 2 3]	[1 2 3]	[1 2 3]
h (original)	[0 1 0.5]	[0 1 0.5]	[0 1 0.5]	[0 1 0.5]	[0 1 0.5]	[0 1 0.5]	[0 1 0.5]
h (flipped)	[0.5 1 0]	[0.5 1 0]	[0.5 1 0]	[0.5 1 0]	[0.5 1 0]	[0.5 1 0]	[0.5 1 0]
y (partial)	[]	[0]	[0 1]	[0 1 2.5]	[0 1 2.5 4]	[0 1 2.5 4 1.5]	[0 1 2.5 4 1.5]

Key Moments - 3 Insights

Why do we flip the second signal h before shifting?

Flipping h reverses it in time, which is essential for convolution to combine signals correctly, as shown in the 'h values (flipped and shifted)' column in the execution_table.

Why does the output y have length length(x)+length(h)-1?

Because convolution sums over all overlapping positions of x and shifted h, including partial overlaps at the edges, as seen in the execution_table steps 1 to 5.

What happens when parts of h extend beyond x during shifting?

Those parts multiply with zero (no overlap), contributing zero to the sum, as in step 1 where h overlaps with only the first element of x.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3. What is the sum at this step?

A 1.5

B 3.0

C 2.5

D 4.0

💡 Hint

Check the 'Sum at step' column in row for Step 3 in execution_table.

At which step does the convolution output first become non-zero?

A Step 2

B Step 1

C Step 3

D Step 4

💡 Hint

Look at the 'y output' column in execution_table; first non-zero value appears at Step 2.

If the vector h was longer, how would the length of y change?

A It would stay the same

B It would increase

C It would decrease

D It would become zero

💡 Hint

Recall the exit_note and the formula length(y) = length(x) + length(h) - 1.

Concept Snapshot

Convolution combines two signals by flipping one,
shifting it across the other, multiplying overlapping values,
and summing these products at each shift.
In MATLAB, use y = conv(x, h) to compute this.
Output length is length(x)+length(h)-1.

Full Transcript

Convolution is a way to combine two signals by flipping one signal, shifting it step by step over the other, multiplying the overlapping values, and summing these products to get each output value. In MATLAB, the conv function does this automatically. The output vector length is the sum of the input lengths minus one. This process is shown step-by-step in the execution table, where at each step we multiply overlapping elements and sum them to build the output. Beginners often wonder why we flip the second signal and why the output length is longer than inputs; these are explained by the convolution definition and the overlapping process.

Practice

1. What does the MATLAB function conv(x, h) compute?

easy

A. The convolution of vectors x and h

B. The element-wise multiplication of x and h

C. The dot product of x and h

D. The sum of vectors x and h

Solution

1. Step 1: Understand the function purpose

   The conv function in MATLAB is designed to compute the convolution of two vectors.

2. Step 2: Differentiate convolution from other operations

   Convolution combines two signals to produce a third, different from element-wise multiplication, dot product, or sum.

3. Final Answer:

   The convolution of vectors x and h -> Option A

4. Quick Check:

   conv(x, h) = convolution [OK]

Hint: Remember conv() means convolution, not multiplication [OK]

Common Mistakes:

• Confusing convolution with element-wise multiplication
• Thinking conv() sums vectors
• Mixing up dot product with convolution

2. Which of the following is the correct syntax to compute convolution of vectors a and b in MATLAB?

easy

A. conv(a, b)

B. convolve(a, b)

C. a * b

D. conv(a + b)

Solution

1. Step 1: Recall MATLAB convolution syntax

   MATLAB uses the function conv(x, h) to compute convolution of two vectors.

2. Step 2: Check each option

   conv(a, b) matches the correct syntax. convolve(a, b) is not a MATLAB function. a * b is matrix multiplication, not convolution. conv(a + b) has wrong argument format.

3. Final Answer:

   conv(a, b) -> Option A

4. Quick Check:

   Use conv(x, h) syntax [OK]

Hint: Use conv() with two arguments separated by comma [OK]

Common Mistakes:

• Using non-existent function convolve()
• Trying to multiply vectors with * for convolution
• Passing only one argument to conv()

3. What is the length of the output vector when you compute conv([1 2 3], [4 5]) in MATLAB?

medium

A. 6

B. 3

C. 5

D. 4

Solution

1. Step 1: Identify lengths of input vectors

   Vector [1 2 3] has length 3, and [4 5] has length 2.

2. Step 2: Calculate output length of convolution

   Output length = length(x) + length(h) - 1 = 3 + 2 - 1 = 4.

3. Final Answer:

   4 -> Option D

4. Quick Check:

   Output length = 3 + 2 - 1 = 4 [OK]

Hint: Output length = sum of input lengths minus one [OK]

Common Mistakes:

• Assuming output length equals input length
• Adding lengths without subtracting one
• Confusing convolution length with element-wise length

4. Consider the MATLAB code:

a = [1 2]; b = [3 4 5]; c = conv(a b);

What is the error in this code?

medium

A. Vectors a and b have different lengths

B. Missing comma between a and b in conv function

C. conv function does not exist in MATLAB

D. Using square brackets instead of parentheses

Solution

1. Step 1: Check syntax of conv function call

   The function call conv(a b) is missing a comma between the two input vectors.

2. Step 2: Verify other options

   Vectors can have different lengths for convolution. conv() is a valid MATLAB function. Square brackets are used correctly for vector definition, not in function call.

3. Final Answer:

   Missing comma between a and b in conv function -> Option B

4. Quick Check:

   Use comma to separate arguments in conv() [OK]

Hint: Separate arguments with comma inside conv() [OK]

Common Mistakes:

• Omitting comma between function arguments
• Thinking vectors must be same length
• Confusing vector definition brackets with function call syntax

5. You want to convolve two signals x = [1 0 2] and h = [3 4] but only keep the output length equal to the length of x. Which MATLAB command achieves this?

hard

A. conv(x, h, 'full')

B. conv(x, h)

C. conv(x, h, 'same')

D. conv(x, h, 'valid')

Solution

1. Step 1: Understand convolution output length options

   By default, conv(x, h) returns full convolution with length length(x)+length(h)-1.

2. Step 2: Use 'same' option to keep output length equal to x

   Using conv(x, h, 'same') returns the central part of the convolution, matching length of x.

3. Final Answer:

   conv(x, h, 'same') -> Option C

4. Quick Check:

   'same' option keeps output length equal to first input [OK]

Hint: Use 'same' option in conv() to match input length [OK]

Common Mistakes:

• Using default conv() which returns longer output
• Confusing 'valid' and 'same' options
• Expecting conv() to truncate output automatically