1 Crore+ students have signed up on EduRev. Have you? 
DFT provides an alternative approach to time domain convolution. It can be used to perform linear filtering in frequency domain.
Thus,
The problem in this frequency domain approach is that Y(ω), X(ω) and H(ω) are continuous function of ω, which is not fruitful for digital computation on computers. However, DFT provides sampled version of these waveforms to solve the purpose.
The advantage is that, having knowledge of faster DFT techniques likes of FFT, a computationally higher efficient algorithm can be developed for digital computer computation in comparison with time domain approach.
Consider a finite duration sequence, [x(n) = 0, for, n < 0 and n ≥ L] (generalized equation), excites a linear filter with impulse response [h (n) = 0,for n < 0 and n ≥ M].
From the convolution analysis, it is clear that, the duration of y(n) is L+M−1.
In frequency domain
Now, Y(ω) is a continuous function of ω and it is sampled at a set of discrete frequencies with number of distinct samples which must be equal to or exceeds L+M−1.
Y(ω) = X(k).H(k), where k = 0,1,….,N1
Where, X(k) and H(k) are Npoint DFTs of x(n) and h(n) respectively. x(n)&h(n) are padded with zeros up to the length N. It will not distort the continuous spectra X(ω) and H(ω). Since N ≥ L + M − 1, Npoint DFT of output sequence y(n) is sufficient to represent y(n) in frequency domain and these facts infer that the multiplication of Npoint DFTs of X(k) and H(k), followed by the computation of Npoint IDFT must yield y(n).
This implies, Npoint circular convolution of x(n) and H(n) with zero padding, equals to linear convolution of x(n) and h(n).
Thus, DFT can be used for linear filtering.
Caution − N should always be greater than or equal to L+M−1. Otherwise, aliasing effect would corrupt the output sequence.
32 videos76 docs64 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
32 videos76 docs64 tests
