EE 624 Advanced DSP

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EE 624 Advanced DSP

• Lecture Notes 1

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Review of Signals Concepts

• Fourier analysis tells us about the frequency
  content of a signal (frequency spectrum) and also
  the energy in the signal.
• Analysis of digital signals tends to mimic
  analysis of analog signals. We just make
  adjustments based on the sampling process.

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Fourier Series

• Periodic waveforms can be represented by a
  Fourier Series
• ?0 fundamental frequency
• Tp Period of waveform

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Fourier Series Coefficients
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Complex Fourier Series Form
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Continuous Time Fourier Transform

• What happens as the period of the periodic wave
  gets longer (Tp gets larger?
• Fundamental frequency gets lower
• Harmonics get closer together
• In the limit, as Tp increases to ?, the spacing
  between harmonics goes to d?/2?. (1/Tp ?/2?
  with Tp ? ?.) That is, the frequency spectrum
  becomes continuous.
• The discrete frequency coefficients become a
  continuous function. dn ? d(?).

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Continuous Time Fourier Transform

• Thus, the Fourier Series becomes the Fourier
  Transform for nonperiodic signals (i.e. a
  periodic signal with infinite period).

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Continuous Signals

• Fourier Series ? Function is continuous in time
  and periodic.
• Fourier Transform ? Function is continuous in
  time and not periodic.

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Example Rectangular Pulse
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Example Rectangular Pulse
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Example Rectangular Pulse
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Discrete Time Fourier Transform (DTFT)

• If the (nonperiodic) function is discrete in time
  instead of continuous we must alter the
  Continuous Fourier Transform formula.
• dt is no longer infinitesimally small and f(t) is
  not continuous so the integral becomes a
  summation of discrete values.

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Discrete Time Fourier Transform (DTFT)
T is generally the period between the discrete
values (the sample period). In more common
notation (to not confuse function f with
frequency f)
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Discrete Fourier Transform (DFT)

• In a real world computerized system though we are
  not really considering this exact type of
  function.
• We are not sampling for all time (not an infinite
  series).
• We begin sampling at time t0 and limit ourselves
  to a number of samples, N

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Discrete Fourier Transform (DFT)
The sampled data sequence is
We can then transform this time sequence to a
discrete number of frequency components (in the
interval 0 to 2? radians).
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Discrete Fourier Transform (DFT)

• Modifying the DTFT to a finite number of samples
  and frequency components we have

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Discrete Signals

• Discrete Time Fourier Transform (DTFT) ? Applied
  to an infinite sequence of data which has
  continuous (infinite) frequencies.
• Discrete Fourier Transform (DFT) ? Computes a
  finite number of frequency components for a
  finite sequence of data.

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Inverse DFT (IDFT)

• The IDFT, or synthesis, equation is then given as

This is analagous to the inverse Fourier
transform for continuous time signals
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DFT Frequency Components
What really is O? Instead of continuous
frequencies ?, we are counting discrete
frequencies kO.
We are simply dividing up the frequency range
between 0 and ?s into N regions. Its between 0
and ?s because k 0, 1, 2, , N-1