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The linear convolution of two signals $$x(n)$$ and $$h(n)$$ is given by We will also see that the inverse DFT of the product of the DFT of two signals corresponds to a time-domain operation called the circular convolution. Then, after pointing out some observations about the linear convolution and the DFT, we will see how the DFT can be used to perform the linear convolution. In this article, we will briefly review the linear convolution. For a filter longer than nearly 64 taps, the DFT-based method would be computationally more efficient than the direct- or cascade-form structures (see the last section of chapter 18 of this book). Hence, the DFT-based method can be particularly helpful in implementing an FIR filter. Note that the difference equation of a Finite Impulse Response (FIR) filter, repeated below for the convenience, is actually calculating the convolution of the input sequence, $$x(n)$$, with the filter coefficients $$b_k$$:
![convolution fourier transform convolution fourier transform](https://i.ytimg.com/vi/ZrMff7yGfqo/maxresdefault.jpg)
However, since there are efficient methods of calculating the DFT, collectively called the Fast Fourier Transform (FFT), we can gain significant computational saving by using the DFT to perform the time-domain convolution.
![convolution fourier transform convolution fourier transform](https://image3.slideserve.com/6359219/example2-l.jpg)
You may doubt the efficiency of this method because we are replacing the convolution operation with two DFTs, one multiplication, and an inverse DFT operation. This can be achieved by multiplying the DFT representation of the two signals and then calculating the inverse DFT of the result. One of the most important applications of the Discrete Fourier Transform (DFT) is calculating the time-domain convolution of signals. The DFT provides an efficient way to calculate the time-domain convolution of two signals.
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