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Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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FIR filters can be useful in making computer-aided design of the filters. Let us take an example and see how it works. Given below is a figure of desired filter.

Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

While doing computer designing, we break the whole continuous graph figures into discrete values. Within certain limits, we break it into either 64, 256 or 512 (and so on) number of parts having discrete magnitudes.

In the above example, we have taken limits between -π to +π. We have divided it into 256 parts. The points can be represented as H(0), H(1),….up to H(256). Here, we apply IDFT algorithm and this will give us linear phase characteristics.

Sometimes, we may be interested in some particular order of filter. Let us say we want to realize the above given design through 9th order filter. So, we take filter values as h0, h1, h2….h9. Mathematically, it can be shown as below

Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Where there are large number of dislocations, we take maximum points.

For example, in the above figure, there is a sudden drop of slopping between the points B and C. So, we try to take more discrete values at this point, but there is a constant slope between point C and D. There we take less number of discrete values.

For designing the above filter, we go through minimization process as follows;

Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Similarly,

Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Representing the above equation in matrix form, we have −

Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Let us take the 1000×1 matrix as B, 1000×9 matrix as A and 9×1 matrix as  Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

So, for solving the above matrix, we will write

  Computer Aided Design - Fast Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 

where A* represents the complex conjugate of the matrix A.

The document Computer Aided Design - Fast Fourier Transform 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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