Compare linear convolution and circular convolution

Linear convolution and circular convolution are two methods used in signal processing to combine two signals or sequences. They differ in how they handle signal boundary conditions and their application contexts. Here's a comparison between linear convolution and circular convolution:

Linear Convolution:

1. Boundary Conditions:

   • Finite-Length Signals: Linear convolution is typically applied to finite-length signals where the signals are assumed to be zero outside their defined intervals.
   • Zero-Padding: Often involves zero-padding of signals to handle the convolution process efficiently and avoid boundary effects.

2. Mathematical Definition:

   • Linear convolution y[n]=x[n]∗h[n]y[n] = x[n] \ast h[n]y[n]=x[n]∗h[n] of two discrete-time signals x[n]x[n]x[n] and h[n]h[n]h[n] is defined as: y[n]=∑k=−∞∞x[k]⋅h[n−k]y[n] = \sum_{k=-\infty}^{\infty} x[k] \cdot h[n-k]y[n]=∑k=−∞∞?x[k]⋅h[n−k]
   • This operation computes the sum of products of overlapping segments of the signals, reflecting the linear combination of the sequences.

3. Application:

   • Widely used in digital signal processing (DSP), filtering operations, system analysis, and real-world applications where signals are finite and non-periodic.

4. Output Length:

   • The output length of linear convolution is typically longer than the input signals due to overlap and extension effects, unless specific techniques like truncation or overlap-add/save are applied.

Circular Convolution:

1. Boundary Conditions:

   • Periodic Signals: Circular convolution assumes periodicity in signals, meaning the signal values repeat indefinitely.
   • No Zero-Padding: There's no need for zero-padding since the signals are considered periodic.

2. Mathematical Definition:

   • Circular convolution y[n]=x[n]?h[n]y[n] = x[n] \circledast h[n]y[n]=x[n]?h[n] of two discrete-time signals x[n]x[n]x[n] and h[n]h[n]h[n] is defined as: y[n]=∑k=0N−1x[k]⋅h[(n−k)mod  N]y[n] = \sum_{k=0}^{N-1} x[k] \cdot h[(n-k) \mod N]y[n]=∑k=0N−1?x[k]⋅h[(n−k)modN] where NNN is the period of the signals.
   • This operation effectively wraps the signals around at their boundaries, simulating a circular arrangement.

3. Application:

   • Commonly used in applications where signals exhibit periodic characteristics, such as in communication systems (e.g., modulation/demodulation processes) and in spectral analysis using the Discrete Fourier Transform (DFT).

4. Output Length:

   • The output length of circular convolution is typically the same as the input signals, as it assumes periodicity and wraps around the signal boundaries.

Key Differences:

• Boundary Effects: Linear convolution handles signals with finite extent and uses zero-padding to manage boundary effects, while circular convolution assumes periodicity and wraps signals around cyclically.

• Output Characteristics: Linear convolution typically results in an output sequence longer than the input sequences due to overlap and extension, whereas circular convolution preserves the input signal length.

• Applications: Linear convolution is more applicable in general signal processing tasks, while circular convolution is specialized for periodic signals and spectral analysis.

In summary, understanding the differences between linear and circular convolution helps in selecting the appropriate method based on the characteristics of the signals and the specific requirements of the application in signal processing and related fields.