Figure 1. Figure 2. The linear convolution between the two delays x [ n ] x[n] x [ n ] by one sample, as expected. Figure 3. Linear convolution between x [ n ] x[n] x [ n ] and h [ n ] h[n] h [ n ].
However, the circular convolution performs a circular shift of the signal x [ n ] x[n] x [ n ]. Figure 4. Circular convolution between x [ n ] x[n] x [ n ] and h [ n ] h[n] h [ n ]. Equation 9 is identical to Equation 1, i. While the DFS assumes that the signal is periodic, i. It means we can state it but we cannot enforce it. The only solution to that, would be padding the signal vector x [ n ] x[n] x [ n ] with infinitely many zeros. Figure 5. Figure 6. Figure 7. This yields.
Figure 8. Magnitude discrete-frequency coefficients of x [ n ] x[n] x [ n ]. Figure 9. Magnitude DFT of x [ n ] x[n] x [ n ] naively visualized with zeros surrounding the 4 nonzero coefficients.
Figure True magnitude DFT of x [ n ] x[n] x [ n ]. Discrete-time Fourier transform is the z-transform evaluated on the unit circle [2]. Since we sample around a circle, after N N N samples we wrap around and start sampling the same points again. Having established that the DFT is periodic, we can now explain the circular convolution phenomenon.
I like to think of it as aliasing in the time domain. Note: We have discussed the notion of aliasing in the frequency domain in one of the previous articles. In Figure 3 we can see that the linear convolution between x x x and h h h is of length 5. We achieve it by padding h h h with 2 zeros, i. Now x x x and h h h are of equal length and their DFTs are as well.
If x x x and h h h were continuous-time and we were using the Fourier transform instead of the discrete Fourier transform, the convolution theorem would tell us that y y y is the convolution of x x x and h h h. We explicitly showed that the convolution of x x x and h h h should be a discrete signal of length 5 see Figure 3. How long is y y y? Inverse DFT inherently assumes that the time domain signal is of the same length as the frequency-domain coefficient vector.
Thus, we introduced aliasing in the time domain; hence the wrap-around of the last sample in Figure 4, and more broadly, circular convolution effect. We have seen that the circular convolution somehow distorts the linear convolution. But in our example, x x x was circularly shifted, not completetely destroyed. Do it graphically and verify your results by multiplying the DTFTs of x[n] and y[n]. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.
Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. The circular convolution of the zero-padded vectors, xpad and ypad , is equivalent to the linear convolution of x and y.
The DFT is one of the most powerful tools in digital signal processing which enables us to find the spectrum of a finite-duration signal. There are many circumstances in which we need to determine the frequency content of a time-domain signal.
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