The fft and frequency domain representations a for the eight...

The FFT and Frequency-Domain Representations a. For the eight-point decimation-in-time (DIT) flow graph below: • What is the signal path gain from input x[7] to output X[2]? • How many paths lead from input x[7] to output X[2]? Is this true for each output? • Use the flow graph to write the expression for X[3].

Transcript

00:01 Hi, in this question we have to develop a radix to dif that is the decimation in frequency.

00:08 So in ftt algorithm, it is given by fast fourier transform.

00:16 So here for n is equal to 8.

00:19 Now we have to so first we have to arrange the sequence xn in bit reversed order.

00:29 So we have it is so order it is given by it is x of 0, x of 4, so x of 2, x of 6, x of 1, x of 5 and x of 3 and x of 7.

00:50 Then we have to perform n by 2 that is 4 stage of butterfly butterfly computation where we have so for the first case stage 1 that is butterfly with w power 0.

01:14 So it can be given by it is so compute so compute x of 0 is equal to x of 0 plus x of 4.

01:25 Similarly, x of 4 is equal to x of 0 minus x of 4 and also x of 2 is given by x of 2 plus x of 6 and similarly for x of 6 we can write it as x of 2 minus x of 2.

01:44 Then we have x of 1 is given by x of 1 plus x of 5.

01:50 So we can write x of 5 as x of 1 minus x of 5.

01:55 Then we have x of 3 can be written as x of 3 plus x of 7.

02:02 Similarly, we can write for x of 7 can be written as x of 3 minus x of 7.

02:08 Now for stage 2 we have the butterfly with omega power 0 and w power 0.

02:17 So we have x of 0 is equal to x of 0 plus x of 2 whereas x of 2 will be equal to x of 0 minus x of 2.

02:26 Then we have x of 4 is equal to x of 4 plus x of 6.

02:31 Then x of 6 will be equal to so it is x of 4 minus x of 6.

02:36 Then so here minus so the same term.

02:40 Then we have x of 1 is equal to x of 1 plus x of 3.

02:44 Then x of 3 will be equal to x of 1 minus x of 3.

02:48 X of 5 will be equal to x of 5 plus x of 7.