SIGNAL PROCESSING

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SIGNAL PROCESSING

• What is signal processing ?
• The operation done on a signal in some fashion
  to extract some useful information.

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FFT USING MATLAB

• What is FFT?
• A fast Fourier transform (FFT) is an efficient
  algorithm to compute the discrete Fourier
  transform (DFT) and its inverse.

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DFT

• What is DFT?
• It is a kind of Fourier transform which
  transforms one function into another.
• DFT requires an input function that is discrete
  and whose non-zero values have a limited (finite)
  duration.

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fft - Discrete Fourier transform

• Syntax
• Y fft(X)Y fft(X,n)Y fft(X,,dim)Y
  fft(X,n,dim)

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Y fft(X)

• Y fft(X) returns the discrete Fourier
  transform (DFT) of vector X, computed with a fast
  Fourier transform (FFT) algorithm.

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Yfft(X,n)

• Y fft(X,n) returns the n-point DFT. If the
  length of X is less than n, X is padded with
  trailing zeros to length n. If the length of X is
  greater than n, the sequence X is truncated. When
  X is a matrix, the length of the columns are
  adjusted in the same manner.

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Y fft(X,,dim)

• Y fft(X,,dim) and Y fft(X,n,dim) applies
  the FFT operation across the dimension dim.

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EXAMPLES

• A common use of Fourier transforms is to find
  the frequency components of a signal buried in a
  noisy time domain signal

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• Consider data sampled at 1000 Hz. Form a signal
  containing a 50 Hz sinusoid of amplitude 0.7 and
  120 Hz sinusoid of amplitude 1 and corrupt it
  with some zero-mean random noise

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PROGRAM

• Fs 1000
• T 1/Fs
• L 1000
• t (0L-1)T
• x 0.7sin(2pi50t) sin(2pi120t)
• y x 2randn(size(t))
• plot (Fst(150),y(150))
• title ('Signal Corrupted with Zero-Mean Random
  Noise')
• xlabel ('time (milliseconds)')

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• It is difficult to identify the frequency
  components by looking at the original signal.
  Converting to the frequency domain, the discrete
  Fourier transform of the noisy signal y is found
  by taking the fast Fourier transform (FFT)

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Program

• NFFT 2nextpow2(L)
• Y fft(y,NFFT)/L
• f Fs/2linspace(0,1,NFFT/21)
• Plot single-sided amplitude spectrum.
• plot(f,2abs(Y(1NFFT/21)))
• title ('Single-Sided Amplitude Spectrum of y(t)')
• xlabel ('Frequency (Hz)')
• ylabel('Y(f)')

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• The main reason the amplitudes are not exactly
  at 0.7 and 1 is because of the noise. Several
  executions of this code (including recomputation
  of y) will produce different approximations to
  0.7 and 1. The other reason is that you have a
  finite length signal. Increasing L from 1000 to
  10000 in the example above will produce much
  better approximations on average.

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CONVOLUTION

• Syntax
• w conv(u,v)
• w conv(u,v) convolves vectors u and v.
  Algebraically, convolution is the same operation
  as multiplying the polynomials whose coefficients
  are the elements of u and v.

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• C conv(...,'shape')
• C conv(...,'shape') returns a subsection of the
  convolution, as specified by the shape parameter
  
• full returns the full convolution (default).
• same returns the central part of the convolution
  of the same size as A.
• valid Returns only those parts of the convolution
  that are computed without the zero-padded edges.

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• enter the 1st sequence 1,2,3,4,5,6
• enter the 2st sequence 2,5,3,4,6,7
• the resultant signal is
• y 2 9 19 33 53 80 93 83
  82 71 42

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