Real time DSP

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Real time DSP

• Professors
• Eng. Julian S. Bruno
• Eng. Jerónimo F. Atencio
• Sr. Lucio Martinez Garbino

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Discrete Time Fourier Transform
DTFT
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Discrete Fourier Series (I)
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Discrete Fourier Series (II)
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Discrete Fourier Series (III)
DTFT of periodical signal
DFS coefficients
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Discrete Fourier Transform (I)
xm is an aperiodic sequence
Sampling ?k2pk/N
DFS
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Discrete Fourier Transform (II)
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Discrete Fourier Transform (III)
0 k N-1
0 n N-1
The inherent periodicity is always present
Propierties Circular Shift of a Sequence
Circular Convolution
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Sampling the Fourier Transform
DTFT
Sampling
DFT
DFS
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DFT Propierties
Circular Shift of a Sequence
Circular Convolution
The circular convolution corresponding to
X1kX2k is identical to the linear convolution
corresponding to X1(ejw)X2(ejw) if N, the length
of the DFTs, satisfies N L P - 1 .
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Implementing Linear Time-Invariant Systems Using
the DFT
xn
yn xn hn
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Overlap-addmethod
yrn xrn hn
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Overlap-savemethod
yrn xrn hn
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Understanding the DFT Equation

• In this example we have a 4 samples signal and we
  use DFT to get its frequency representation.
• The result for each frequency component is
  obtained after computing 8 real sums and
  multiplications.

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DFT example (I)

• Consider a signal formed with 2 sinusoidal, one
  of 1 KHz and the other of 2 KHz and a phase shift
  of ¾p.
• N 8 samples.
• Fs 8000 samples/s.
• Fs/N 1Khz
• First computations are showed in detail.

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DFT example (II)
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DFT example (III)
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DFT example (IV)

• Here we show the final result in both
  representations formats.
• The complex DFT outputs for m1 to m(N/2)-1 are
  redundant with frequency output values form
  mgt(N/2)
• We can see an even symmetry in Magnitude and Real
  representations, while an odd symmetry in
  Imaginary and Phase.
• It can be verified the amplitude and phase
  relationship between the sinusoidal components,
  but absolute values?

Fixed point DSP
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DFT Leakage (I)
Leakage is an unavoidable fact of life when we
perform the DFT on real world finite-length time
sequences

• If there are frequency components that are not
  integer multiples of fres, we got leakage.
• Leakage evidences the effect of sampling during
  finite (and rectangular) time window.

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DFT Leakage (II)

• As can be seen, the sinc function is always
  present, but only evidenced when frequency
  components are not integer multiples of fres.
• The DFT output is a sampled version of the
  continuous spectral

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Time Windowing (I)

• The only fact of considering a finite length time
  sequence, is equivalent to convolve a sinc with
  all frequency samples.
• If we use a window, we will convolve with the
  spectrum this window.
• The net effect of windowing is a better spectral
  estimation, reducing leakage and picket fence
  effect.

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Time Windowing (II)

• Spectral analysis
• Equivalent Noise Bandwith
• Processing Gain
• Overlap Correlation
• Scalloping Loos
• Worst Case Processing Loss
• Minimun Resolution Bandwidth

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Equivalent Noise Bandwith
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Processing Gain
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Overlap Correlation
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Picket fence effect

• The picket fence effect is a manifestation of
  applying DFT over a finite time sequence.
• The net effect of windowing is a smoothed
  frequency response of a sinc at each frequency
  index.

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Minimun Resolution Bandwidth
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Time Windowing (II)
The window selection is a trade-off between main
lobe widening, first sidelobe levels, and how
fast the sidelobes decrease with increased
frequency.
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Goertzel Algorithm
The Goertzel algorithm is a digital signal
processing technique for identifying frequency
components of a signal, published by Dr. Gerald
Goertzel in 1958
If you implement the Goertzel algorithm L times
to detect L different tones, Goertzel is more
efficent than FFT when Llt log2N
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Goertzel Algorithm Implementation
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Zoom FFT
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Zero Stuffing

• Zero stuffing is a way of increasing frequency
  resolution.
• The spectrum visualized corresponds to the
  convolution of a sinusoidal and a rectangular
  signal.
• Thus, the underlying spectrum of the sinusoidal
  is distorted by a sinc.