Convolution is a fundamental concept in Digital Signal Processing (DSP) used to analyze and modify signals. Linear and circular convolution are two essential methods, but their applications differ significantly.
If you’re wondering:
- When should I use linear vs. circular convolution?
- Why does circular convolution require zero-padding?
- How do their formulas and results differ?
This guide directly compares linear and circular convolution, explains their formulas, and provides a practical example to clarify their differences. Let’s get straight to it.
Update for 2025:
This guide has been reviewed and updated with the latest insights into DSP applications of linear and circular convolution. We’ve included new examples and FAQs to clarify key concepts.
| Point of Comparison | Linear Convolution | Circular Convolution |
| Definition | Linear convolution is a mathematical operation done to calculate the output of any Linear-Time Invariant (LTI) system given its input and impulse response. | Circular convolution is essentially the same process as linear convolution. Just like linear convolution, it involves the operation of folding a sequence, shifting it, multiplying it with another sequence, and summing the resulting products (We’ll see what this means in a minute). However, in circular convolution, the signals are all periodic. Thus the shifting can be thought of as actually being a rotation. Since the values keep repeating because of the periodicity. Hence, it is known as circular convolution. |
| Application | It is applicable for both continuous and discrete-time signals. | Circular convolution is also applicable for both continuous and discrete-time signals. |
| Mathematical Representation | We can represent Linear Convolution as
y(n)=x(n)*h(n) Here, y(n) is the output (also known as convolution sum). x(n) is the input signal, and h(n) is the impulse response of the LTI system. |
We can represent Circular Convolution as
y(n)=x(n)⊕h(n) Here y(n) is a periodic output, x(n) is a periodic input, and h(n) is the periodic impulse response of the LTI system. |
| Sequence Lengths | In linear convolution, both the sequences (input and impulse response) may or may not be of equal sizes. That is, they may or may not have the same number of samples. Thus the output, too, may or may not have the same number of samples as any of the inputs. | In circular convolution, both the sequences (input and impulse response) must be of equal sizes. They must have the same number of samples. Thus the output of a circular convolution has the same number of samples as the two inputs. |
| Example with Sequence Lengths | For example, consider the following signals:
x(n): [1,2,3] h(n): [1,2,3,4,5] As you can see, the number of samples in the input and Impulse response signals is not the same. Still, linear convolution is possible. Here’s how you calculate the number of samples in the output of linear convolution. L = M + N – 1 Where M is the number of samples in x(n). N is the number of samples in h(n). For the above example, the output will have (3+5-1) = 7 samples. |
For the given example, circular convolution is possible only after modifying the signals via a method known as zero padding. In zero padding, zeroes are appended to the sequence that has a lesser size to make the sizes of the two sequences equal. Thus, for the given sequence, after zero-padding:
x(n) = [1,2,3,0,0] Now both x(n) and h(n) have the same lengths. So circular convolution can take place. And the output of the circular convolution will have the same number of samples. i.e., 5. |
| Shifting Process | Graphically, when we perform linear convolution, there is a linear shift taking place. Check out the formula for a convolution.
 x(k)h(n-k)
There is a folding of the IR sequence, shifting it by n, multiplying it with another sequence (input), and summing the resulting products. |
Graphically, when we perform circular convolution, there is a circular shift taking place. Alternatively, we can call it rotation. |
| Filter Response Calculation | It is possible to find the response of a filter using linear convolution. | It is possible to find the response of a filter using circular convolution after zero padding. I fact, we will be doing this in overlap-save and overlap-add methods — two essential topics in our digital signal processing course. |
| Output Characteristics | Linear convolution may or may not result in a periodic output signal. | The output of a circular convolution is always periodic, and its period is specified by the periods of one of its inputs. |
Frequently Asked Questions (FAQs) about Linear and Circular Convolution
What is the difference between linear convolution and circular convolution in DSP?
Linear convolution is used to determine the output of an LTI system, while circular convolution is primarily used in Fourier analysis and periodic signals.
What are the key applications of linear and circular convolution?
Linear convolution is used in system response calculations, while circular convolution is useful in FFT-based signal processing techniques.
Why do we use circular convolution instead of linear convolution?
Circular convolution is often preferred when working with periodic signals or when implementing fast convolution using the Fast Fourier Transform (FFT).
How do you calculate linear and circular convolution?
Linear convolution is calculated using the formula y(n) = x(n) * h(n), while circular convolution is given by y(n) = x(n) ⊕ h(n). Zero-padding is often used to make circular convolution work for unequal-length sequences.
What is the formula for linear and circular convolution?
Linear Convolution: y(n) = \sum x(k) h(n-k)
Circular Convolution: y(n) = \sum x(k) h((n-k) \mod N)
