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EE200_Weber_11-18

# EE200_Weber_11-18 - EE 200 Fourier Transform vs Fourier...

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29 EE 200 Fourier Transform vs. Fourier Series Do we need two types of Fourier operations for both types of signals (continuous and discrete)? Fourier transform: non-periodic signals Fourier series: periodic signals If we can apply the Fourier transform to a non-periodic signal, why can’t we use it for a periodic signal? A non-periodic signal with finite extent can be viewed as one period of a periodic signal. So can we then use the Fourier series to analyze it?

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30 EE 200 Fourier Transform vs. Fourier Series If we have infinite signal, y(n) , we can define a finite length portion of it, y (n) , that is zero outside the domain of y(n) . We can create a periodic signal, x(n) , by repeating it the finite length signal y (n) . We can find the DTFT of the finite length signal, y (n) We can find the DFS of the periodic signal, x(n) y(n) x(n) n n x ( n ) = " y ( n # mp ) m = #\$ \$ % p
31 EE 200 Fourier Transform vs. Fourier Series The discrete-time Fourier transform of the finite signal y (n) is The discrete Fourier series (a.k.a. the DFT) of the periodic signal x(n) is " Y ( # ) = " y ( n ) n = \$% % & e \$ in # = y ( n ) n = \$ 0 p \$ 1 & e \$ in # X k = x ( n ) n = 0 p " 1 # e " ink \$ 0 = y ( n ) n = " 0 p " 1 # e " ink \$ 0

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32 EE 200 Fourier Transform vs. Fourier Series The relationship between the DTFT and DFT is then The results of the DFT are samples of the output of the DTFT at frequencies ω 0 apart. Looking at it the other way, if we find the DFT, we can get a good approximation of the shape of the DTFT. This assumes the frequency spacing between the DFT values is close enough to represent the DTFT. X k = " Y ( k # 0 )
33 EE 200 Fourier Transform vs. Fourier Series Using the DFT to find the frequency components of non-periodic signals is a valuable tool. In many real- world problems, we have non-periodic signals, but are interested in the frequency domain information in small parts of the signal. We can break up a long non-periodic signal into short pieces that only contain the regions we are interested in. By pretending these are each one period of a periodic signal and taking the DFT of each segment, we can calculate, in a finite amount of time, the frequency present in those parts of the signal.

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34 EE 200 Fourier Transform vs. Fourier Series Using DFTs (and similar transforms) to extract frequency information from a signal can also be used with spatial signals. The lines and columns of an image can be viewed as finite length signals that can be converted to frequency domain representation using DFTs. Once in the frequency domain, the frequency coefficients can be altered, quantized, discarded, etc. in order to perform various operations such as enhancement, compression or restoration.
35 EE 200 Fourier Transform vs. Fourier Series Example: The JPEG standard (.jpg files) for compression of digital images divides the images into blocks of 8 × 8 pixels. Each block is then converted to a

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EE200_Weber_11-18 - EE 200 Fourier Transform vs Fourier...

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