# What Are Linear and Circular Convolution?

I have some basic understanding of signals and convolution. As far as I know it shows the similarities of two signals. Could I get some explanation in plain English of:

• what are the linear and circular convolution
• why they are important
• practical situation where they are used
• No, convolution does not show similarity of signals. Perhaps if you could explain what basic understanding you do have of signals and convolution, it might be easier to answer the questions you ask. – Dilip Sarwate Aug 21 '13 at 23:12
• basically convolution is a process to calculate output of a LTI systems because these systems are not vary with time thats why we can not calculate output directly by using y(t) = h(t)x(t). – user5583 Oct 4 '13 at 5:58
• @DilipSarwate, convolution of two signals is correlation with one of the signals turned around. and correlation does show similarities of two signals. so there is something to the OP's understanding, but it is not complete. – robert bristow-johnson Jul 29 '14 at 14:54
• @robertbristow-johnson Correlation also requires conjugation of one of the signals whereas convolution does. not, and so I disagree that your assertion that "convolution of two signals is correlation with one of the signals turned around." And don't bring up the defense that "it works for real-valued signals"! – Dilip Sarwate Mar 29 '19 at 16:00
• yeah, i knew that @DilipSarwate , it's just that so many times we are correlating real data against real data. – robert bristow-johnson Mar 29 '19 at 20:05

• Linear convolution is the basic operation to calculate the output for any linear time invariant system given its input and its impulse response.

• Circular convolution is the same thing but considering that the support of the signal is periodic (as in a circle, hence the name).

Most often it is considered because it is a mathematical consequence of the discrete Fourier transform (or discrete Fourier series to be precise):

• One of the most efficient ways to implement convolution is by doing multiplication in the frequency.
• Sampling in the frequency requires periodicity in the time domain.
• However, due to the mathematical properties of the FFT this results in circular convolution.

The method needs to be properly modified so that linear convolution can be done (e.g. overlap-add method).

I think you mistake convolution for cross-correlation. They have similar forms, but convolution is more general.

The correlation of two signals $f$ and $g$ could be calculated as: $$\text{corr}(f,g)=\int_{-\infty}^{\infty}f(\tau)^*g(t+\tau)d\tau=(f\star(-g))$$ The convolution of the same signals is: $$(f\star g)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau$$

Convolution could be used to calculate the response of an LTI system, and (normalized) cross-correlation could be used for pattern matching: the maxima of the cross-correlation function is at the offset where pattern g is most likely to be situated in the signal f. If you know this offset you could use a similarity measure (such as the euclidean distance) to quantify similarity.

• Why do you say convolution is more general? Aren't they equivalent if you time reflect one of your signals – Rojo Oct 7 '13 at 21:31
• Does $f(\tau)^*g(t+\tau)$ mean complex conjugation of $f(\tau)$ followed by multiplication? The reason for asking is that in the second equation you write $f(\tau)g(t-\tau)$ without any $*$, and complex conjugation is used in correlation but not in convolution. – Dilip Sarwate Oct 7 '13 at 23:41

Convolution is used to find out the output of an LTI system.If the response of the system to the impulse signal is known($h(t)$ or $h(n)$),then the response to any other input to the system can be found out by convolving the input signal with impulse response.

• How does it answer the question? – jojek Nov 25 '14 at 11:50

Convolution can be understood as the effect one signal has on another signal i.e., you pass a signal through a system (defined by its impulse response) then what is the output? This is answered through convolution.

Likewise, Correlation answers the similarities between two signals. The output of correlation can be positive, zero, or negative.

1. zero: the two signals are not correlated.
2. positive: the two signals are correlated, and both the signals tend to change in the same direction.
3. negative: the two signals are correlated, however both the signals tend to change in the opposite direction.

Correlation is used to find the similarities bwletween any to signals(cross correlation in precise). Linear Convolution is used to find d output of any LTI system (eg. by Flip-shift-drag method etc) while circular Convolution is a special case when d given signal is periodic

Linear convolution: For aperiodic and infinite sequence. Circular convolution: For periodic and finite sequence.