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Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu


This is the first week of a two week laboratory that covers the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)methods. The first week will introduce the DFT and associated samplingand windowing effects, while thesecond week will continue the discussion of the DFT and introduce the FFT.

In previous laboratories, we have used the Discrete-Time Fourier Transform (DTFT) extensively foranalyzing signals and linear time-invariant systems.

(DTFT) X ( e j ω ) = n = - x ( n ) e - j ω n
(inverse DTFT) x ( n ) = 1 2 π - π π X ( e j ω ) e j ω n d ω .

While the DTFT is very useful analytically, it usually cannot be exactlyevaluated on a computer because [link] requires an infinite sum and [link] requires the evaluation of an integral.

The discrete Fourier transform (DFT) is a sampled version of the DTFT, hence it is better suited for numerical evaluation on computers.

(DFT) X N ( k ) = n = 0 N - 1 x ( n ) e - j 2 π k n / N
(inverse DFT) x ( n ) = 1 N k = 0 N - 1 X N ( k ) e j 2 π k n / N

Here X N ( k ) is an N point DFT of x ( n ) . Note that X N ( k ) is a function of a discrete integer k , where k ranges from 0 to N - 1 .

In the following sections, we will study the derivation of the DFT from the DTFT,and several DFT implementations. The fastest and most important implementation is known as the fast Fourier transform (FFT). The FFT algorithmis one of the cornerstones of signal processing.

Deriving the dft from the dtft

Truncating the time-domain signal

The DTFT usually cannot be computed exactly because the sum in [link] is infinite. However, the DTFT may be approximately computedby truncating the sum to a finite window. Let w ( n ) be a rectangular window of length N :

w ( n ) = 1 0 n N - 1 0 else .

Then we may define a truncated signal to be

x tr ( n ) = w ( n ) x ( n ) .

The DTFT of x tr ( n ) is given by:

X tr ( e j ω ) = n = - x tr ( n ) e - j ω n = n = 0 N - 1 x ( n ) e - j ω n .

We would like to compute X ( e j ω ) , but as with filter design, the truncation window distortsthe desired frequency characteristics; X ( e j ω ) and X tr ( e j ω ) are generally not equal. To understand the relation between these two DTFT's,we need to convolve in the frequency domain (as we did in designing filters with the truncationtechnique):

X tr ( e j ω ) = 1 2 π - π π X ( e j σ ) W ( e j ( ω - σ ) ) d σ

where W ( e j ω ) is the DTFT of w ( n ) . [link] is the periodic convolution of X ( e j ω ) and W ( e j ω ) . Hence the true DTFT, X ( e j ω ) , is smoothed via convolution with W ( e j ω ) to produce the truncated DTFT, X tr ( e j ω ) .

We can calculate W ( e j ω ) :

W ( e j ω ) = n = - w ( n ) e - j ω n = n = 0 N - 1 e - j ω n = 1 - e - j ω N 1 - e - j ω , for ω 0 , ± 2 π , ... N , for ω = 0 , ± 2 π , ...

For ω 0 , ± 2 π , ... , we have:

W ( e j ω ) = e - j ω N / 2 e - j ω / 2 e j ω N / 2 - e - j ω N / 2 e j ω / 2 - e - j ω / 2 = e - j ω ( N - 1 ) / 2 sin ( ω N / 2 ) sin ( ω / 2 ) .

Notice that the magnitude of this function is similar to sinc ( ω N / 2 ) except that it is periodic in ω with period 2 π .

Frequency sampling

[link] contains a summation over a finite number of terms. However, we can only evaluate [link] for a finite set of frequencies, ω . We must sample in the frequency domain to compute the DTFT on a computer.We can pick any set of frequency points at which to evaluate [link] , but it is particularly useful to uniformly sample ω at N points, in the range [ 0 , 2 π ) . If we substitute

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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Maira Reply
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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