Fast Fourier Transform

In the fast fourier transform, we used the same signal as that of the DFT experiment so as to compare these two algorithms. We concluded that FFT is better and fast than DFT because calculations are easy and in FFT calculations are performed parallely so speed is more and number of complex addition and multiplication is also less compared to DFT.
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FFT1
FFT2
IFFT1
IFFT2

Discrete Fourier Transform

We had to compute a signal's [x(n)]  Fourier Transform and observe the Magitude Spectrum. Two cases were taken, first was a signal with length equal to 4 and second was the first signal with zero padding to make its length 8. DFT was calculated using the C code, run through Linux terminal. It was observed that DFT always give periodic results and increasing the length by zero padding of signal gives better approximation of signal and resolution of spectrum increases.As N increases frequency spacing reduces, approximation error decreases and resolution improves.
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DFT
IDFT

Discrete Convolution and Correlation

In this experiment,we performed linear convolution,circular convolution and linear convolution using circular convolution. We used the mathematical formula of convolution and observed the results. The length of the input signal in linear convolution was L=5 and M=3, the length of the output signal was N=L+M-1. In circular convolution as we increase the  the length of signal, zeros are added to the output signal. The length of the signal was N=max(L,M).For linear convolution using circular convolution N>= L+M-1. Zero padding was observed.We studied aliasing effect.We also performed cross and auto correlation. The second case was the same input but delayed. In auto correlation the value at the center is largest and the signal is symmetric.
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Correlation
Circular
Linear