Notes

Circular convolution: Circular convolution is another way of finding the convolution sum of two input signals. It resembles the linear convolution, except that the sample values of one of the input signals is folded and right shifted before the convolution sum is found. Also note that circular convolution could also be found by taking the DFT of the two input signals and finding the product of the two frequency domain signals. The Inverse DFT of the product would give the output of the signal in the time domain which is the circular convolution output. The two input signals could have been of varying sample lengths. But we take the DFT of higher point, which ever signals levels to.

For eg. If one of the signals is of length 256 and the other spans 51 samples, then we could only take 256 point DFT. So the output of IDFT would be containing 256 samples instead of 306 samples, which follows N1+N2 – 1 where N1 & N2 are the lengths 256 and 51 respectively of the two inputs. Thus the output which should have been 306 samples long is fitted into 256 samples. The 256 points end up being a distorted version of the correct signal. This process is called circular convolution.



THEORY:
Convolution is a mathematical operation used to express the relation between input and output of an LTI system. It relates input, output and impulse response of an LTI system as
y(n)=x(n)∗h(n)
Where,y (n) = output of LTI
x (n) = input of LTI
h (n) = impulse response of LTI
Discrete Convolution
y(n)=x(n)∗h(n)





THEORY:
DFT of a sequence

X [k ] = ∑ x[n]e− j (2π / N )kn
n=0
Where N= Length of sequence.
K= Frequency Coefficient.
n = Samples in time domain.


FFT: -Fast Fourier transform. There are two methods.
1) Decimation in time (DIT ) FFT.
2) Decimation in Frequency (DIF) FFT. Why we need FFT?
The no of multiplications in DFT = N2. The no of Additions in DFT = N (N-1). For FFT.
The no of multiplication = N/2 log 2N. The no of additions = N log2 N.