DTFT, DFT, ZT, and FFT 501
MATLAB Solution
>> fn _ 8 = [1 2 3 4 5 6 7 8];
>> DFT _ fn _ 8 = (fft(fn _ 8))’ % part(a)DFT _ fn _ 8 =
36.0000
-4.0000 - 9.6569i
-4.0000 - 4.0000i
-4.0000 - 1.6569i
-4.0000
-4.0000 + 1.6569i
-4.0000 + 4.0000i
-4.0000 + 9.6569i>> % note that F(3)= -4.0000 - 1.6569i
>> % F(5)= -4.0000 + 1.6569i, observe F(3)=F(5)*, part(b)>> fn _ 4 = [1 2 3 4]; % part(c)
>> DFT _ fn _ 4 = (fft(fn _ 4))’DFT _ fn _ 4 =
10.0000
-2.0000 - 2.0000i
-2.0000
-2.0000 + 2.0000i>> fn _ 8 _ I=fliplr(fn _ 8); % fn _ 8 _ i =[8 7 6 5 4 3 2 1];
>> DFT _ fn _ 8 _ i = (fft(fn _ 8 _ i))’ % part (d)DFT _ fn _ 8 _ i =
36.0000
4.0000 + 9.6569i
4.0000 + 4.0000i
4.0000 + 1.6569i
4.0000
4.0000 - 1.6569i
4.0000 - 4.0000i
4.0000 - 9.6569i>> fn _ 8 _ 0 = [1 2 3 4 0 0 0 0]; % part(e)
>> DFT _ fn _ 8 _ 0 = (fft(fn _ 8 _ 0))’DFT _ fn _ 8 _ 0 =
10.0000
-0.4142 + 7.2426i
-2.0000 - 2.0000i
2.4142 + 1.2426i
-2.0000
2.4142 - 1.2426i
-2.0000 + 2.0000i
-0.4142 - 7.2426i>> % observe that DFT _ fn _ 8 _ 0 = DFT _ fn _ 4 at four points