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بسم الله الرحمن الرحيمبسم الله الرحمن الرحيم

Digital Signal ProcessingDigital Signal Processing

Lecture 14Lecture 14

FFT-Radix-2 Decimation in Frequency

And Radix -4 Algorithm

University of KhartoumUniversity of Khartoum

Department of Electrical and Electronic EngineeringDepartment of Electrical and Electronic Engineering

Diploma/M. Sc. Program in Telecommunication andInformation Systems 2013-2014Diploma/M. Sc. Program in Telecommunication andInformation Systems 2013-2014

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Dr. Iman AbuelMaaly

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•Another important radix-2 FFTalgorithm, called the decimation-in-frequency algorithm, is obtained by usingthe divide-and-conquer approach.(N=LM , M=2 and L=N/2)

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

twosummations•To derive the algorithm, we begin bysplitting the DFT formula into twosummations, one of which involves thesum over the first N/2 data points and thesecond sum involves the last N/2 datapoints.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

•Thus we obtain:

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

n n+N/2n n+N/2

X(k)Now, let us split (decimate) X(k) into the even-and odd-numbered samples. Thus we obtain

WN2 = WN/2where we have used the fact that WN2 = WN/2

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

k2kk2k

k2k+1k2k+1

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

k2kk2k

k2k+1k2k+1

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Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

X(2k) and X(2k+1)•The computational procedure above can berepeated through decimation of the N/2-point DFTs X(2k) and X(2k+1).

v = log2N•The entire process involves v = log2Nstages of decimation, where each stageinvolves N/2 butterflies of the type shownin Figure 2.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

Figure .2 Basic butterfly computation in thedecimation-in-frequency.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

• Consequently, the computation of the N-pointDFT via the decimation-in-frequency FFTrequires (N/2)log2N complex multiplicationsand N log2N complex additions, just as in thedecimation-in-time algorithm.

For illustrative purposes, the eight-pointdecimation-in-frequency algorithm is given inFigure 3.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

ComputationsComputations

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

ComputationsComputations

Figure .3 First stage of the DIF -FFT algorithm.Figure .3 First stage of the DIF -FFT algorithm.

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Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

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if n =0

x(0) , x(4)

-1-1

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Figure .4 N = 8-piont decimation-in-frequency FFT algorithm.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

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X(0)

X(4)

X(2)

X(6)

X(1)

X(5)

X(3)

X(7)

x(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(7)

x(3)

•We observe from Figure .4 that the input datax(n) occurs in natural order, but the output DFTX(k)occurs in bit-reversed order.

•We also note that the computations areperformed in place.

•However, it is possible to reconfigure thedecimation-in-frequency algorithm so that theinput sequence occurs in bit-reversed orderwhile the output DFT occurs in normal order.

Radix-2 FFT -Decimation in FrequencyRadix-2 FFT -Decimation in Frequency

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

•When the number of data points N in the DFT isa power of 4 (i.e., N = 4v), we can, of course,always use a radix-2 algorithm for thecomputation.

•However, for this case, it is more efficientcomputationally to employ a radix-4 FFTalgorithm.

Radix-4 FFT Algorithm --Radix-4 FFT Algorithm --

L=4M=N/4•Let us begin by describing a radix-4 decimation-in-time FFT algorithm which is obtained by selectingL=4 and M=N/4 in the divide and conquer approach.

LM•For this choice of L and M, we have

l.p = 0, 1, 2, 3; l.p = 0, 1, 2, 3;

m,q = 0, 1, 2, . . , N/4-1m,q = 0, 1, 2, . . , N/4-1;

n = 4m +ln = 4m +l

k = (N/4)p +q k = (N/4)p +q

•We split or decimate the N-point input sequence intofour subsequences,

x(4n), x(4n+1), x(4n+2), x(4n+3), n = 0, 1, ... , N/4-1.x(4n), x(4n+1), x(4n+2), x(4n+3), n = 0, 1, ... , N/4-1.

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

By applying

(1) (1)

We obtain

(2) (2)

where

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

And

The expression in (2) for combining N/4 DFTsdefines Radix-4 decimation in time butterfly whichcan be expressed in matrix form as in equation(3)

Radix-4 FFT Algorithm Radix-4 FFT Algorithm

(3)(3)

Radix-4 FFT AlgorithmRadix-4 FFT Algorithm

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FFT ApplicationsFFT Applications

The Fast Fourier Transform (FFT) is one ofthe most important tools in Digital SignalProcessing.

First, the FFT can calculate a signal'sfrequency spectrum. This is a directexamination of information encoded in thefrequency, phase, and amplitude of thecomponent sinusoids.

•Second, the FFT can find a system's frequencyresponse from the system's impulse response, andvice versa. This allows systems to be analyzed inthe frequency domain, just as convolution allowssystems to be analyzed in the time domain.

•Third, the FFT can be used as an intermediatestep in more elaborate signal processingtechniques. The classic example of this is FFTconvolution, an algorithm for convolving signalsthat is hundreds of times faster than conventionalmethods.

FFT ApplicationsFFT Applications

Use of the DFT in Linear FilteringUse of the DFT in Linear Filtering

•Suppose we have a finite duration sequencex(n) of length L which excites an FIR filter oflength M.

Let

•x(n) =0 for n<0 and n ≥ L

•h(n) =0, for n<0 and n ≥ M

•h(n) is the impulse response of the FIR systemand x(n) is the input to the FIR system.

h(n)

•The output is y(n) with duration L+M-1

•Y(ω) =X(ω) H(ω)

•The DFT of size N ≥ L+M-1 is required torepresent y(n) in the frequency domain.

•But x(n) and h(n) have durations less than N,we pad these sequences with zeros to increasetheir length to N.

•After that we can compute the N-point circularconvolution of x(n) and h(n) to compute y(n).

•This way we can obtain the same results aswould have been obtained with linearconvolution.

FFT Based FilteringFFT Based Filtering

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•Zero-padding is an operation in which morezeros are appended to the original sequence.The resulting longer DFT provides closelyspaced samples of the discrete-times Fouriertransform of the original sequence.

Zero -paddingZero -padding

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•The zero-padding gives us a high-densityspectrum and provides a better displayedversion for plotting. But it does not give us ahigh-resolution spectrum because no newinformation is added to the signal; onlyadditional zeros are added in the data.

•To get high-resolution spectrum, one has toobtain more data from the experiment orobservations.

Zero -paddingZero -padding

Block ConvolutionBlock Convolution

•Segment the infinite-length input sequence intosmaller sections (or blocks), process eachsection using the DFT, and finally assemble theoutput sequence from the outputs of eachsection. This procedure is called a blockconvolution operation.

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تم بحمد الله تعالى