Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Document Description: Linear Filtering for Electronics and Communication Engineering (ECE) 2022 is part of Signals and Systems preparation. The notes and questions for Linear Filtering have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Linear Filtering covers topics like and Linear Filtering Example, for Electronics and Communication Engineering (ECE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Linear Filtering.

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DFT provides an alternative approach to time domain convolution. It can be used to perform linear filtering in frequency domain.

Thus,  Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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].

Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

From the convolution analysis, it is clear that, the duration of y(n) is L+M−1.

In frequency domain

Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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.

Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Y(ω) = X(k).H(k), where k = 0,1,….,N-1

Where, X(k) and H(k) are N-point 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, N-point DFT of output sequence y(n) is sufficient to represent y(n) in frequency domain and these facts infer that the multiplication of N-point DFTs of X(k) and H(k), followed by the computation of N-point IDFT must yield y(n).

This implies, N-point 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.

The document Linear Filtering Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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