Professor A G Constantinides 1 Discrete Fourier Transforms Consider finite duration signal Its z-tranform is Evaluate at points on z-plane as We can evaluate N independent points

Professor A G Constantinides 2 Discrete Fourier Transforms This is known as the Discrete Fourier Transform (DFT) of Periodic in k ie This is as expected since the spectrum is periodic in frequency

Professor A G Constantinides 3 Discrete Fourier Transforms Multiply both sides of the DFT by And add over the frequency index k From which

Professor A G Constantinides 4 Discrete Fourier Transforms This is the inverse DFT That is a) the DFT assumes that we deal with periodic signals in the time domain b) Sampling in one domain produces periodic behaviour in the other domain

Professor A G Constantinides 5 Discrete Fourier Transforms Effectively by knowing is known everywhere since or

Professor A G Constantinides 6 Discrete Fourier Transforms The formula This is essentially an interpolation and forms the basis of the Frequency Sampling Method for FIR digital filter design

Professor A G Constantinides 7 Convolution in DFT Consider the following transform pairs Define Find

Professor A G Constantinides 8 Convolution in DFT From IDFT However

Professor A G Constantinides 9 Convolution in DFT Or Thus This the Circular Convolution

Professor A G Constantinides 10 Computation of the DFT: The FFT Algorithm Computation of DFT requires for every sample N multiplications. There are N samples to be computed i.e. time consuming operations. The Fast Fourier Algorithm:(Decimation in time - DIT, assume even no. of samples) set

Professor A G Constantinides 11 FFT Then DFT of is written set

Professor A G Constantinides 12 FFT ie Or Computations of each of summations is now of order,and thus total computational effort is reduced to. Continuation of “divide-&-compute” reduces effort to Nlog(N)

Professor A G Constantinides 13 8-point FFT 8-point Signal Flow Diagram

Professor A G Constantinides 14 FFT times Time (1 multiplication per microsec) 193 days 4 hrs hr 11 mins sec sec.02 sec64 FFT Direct DFT N

Professor A G Constantinides 15 Decimation-in-Time FFT Algorithm In the basic module two output variables are generated by a weighted combination of two input variables as indicated below where and Basic computational module is called a butterfly computation

Professor A G Constantinides 16 Decimation-in-Time FFT Algorithm Input-output relations of the basic module are: Substituting in the second equation given above we get

Professor A G Constantinides 17 Decimation-in-Time FFT Algorithm Modified butterfly computation requires only one complex multiplication as indicated below Use of the above modified butterfly computation module reduces the total number of complex multiplications by 50%

Professor A G Constantinides 18 Decimation-in-Time FFT Algorithm New flow-graph using the modified butterfly computational module for N = 8

Professor A G Constantinides 19 Decimation-in-Time FFT Algorithm Computational complexity can be reduced further by avoiding multiplications by,,, and The DFT computation algorithm described here also is efficient with regard to memory requirements Note: Each stage employs the same butterfly computation to compute and from and