EE 554454, Fall, 2006 Digital Signal Processing

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EE 554/454, Fall, 2006Digital Signal Processing

• Zhu Han
• Department of Electrical and Computer Engineering
• Class 25
• Dec. 6th, 2007

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Discrete Fourier Transform

• The DFT pair was given as
• Baseline for computational complexity
• Each DFT coefficient requires
• N complex multiplications
• N-1 complex additions
• All N DFT coefficients require
• N2 complex multiplications
• N(N-1) complex additions
• Complexity in terms of real operations
• 4N2 real multiplications
• 2N(N-1) real additions
• Most fast methods are based on symmetry
  properties
• Conjugate symmetry
• Periodicity in n and k

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The Goertzel Algorithm

• Makes use of the periodicity
• Multiply DFT equation with this factor
• Define
• With this definition and using xn0 for nlt0 and
  ngtN-1
• Xk can be viewed as the output of a filter to
  the input xn
• Impulse response of filter
• Xk is the output of the filter at time nN

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The Goertzel Filter

• Goertzel Filter
• Computational complexity
• 4N real multiplications
• 2N real additions
• Slightly less efficient than the direct method
• Multiply both numerator and denominator

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Second Order Goertzel Filter

• Second order Goertzel Filter
• Complexity for one DFT coefficient
• Poles 2N real multiplications and 4N real
  additions
• Zeros Need to be implement only once
• 4 real multiplications and 4 real additions
• Complexity for all DFT coefficients
• Each pole is used for two DFT coefficients
• Approximately N2 real multiplications and 2N2
  real additions
• Do not need to evaluate all N DFT coefficients
• Goertzel Algorithm is more efficient than FFT if
• less than M DFT coefficients are needed
• M lt log2N

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Decimation-In-Time FFT Algorithms

• Makes use of both symmetry and periodicity
• Consider special case of N an integer power of 2
• Separate xn into two sequence of length N/2
• Even indexed samples in the first sequence
• Odd indexed samples in the other sequence
• Substitute variables n2r for n even and n2r1
  for odd
• Gk and Hk are the N/2-point DFTs of each
  subsequence

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Decimation In Time

• 8-point DFT example using decimation-in-time
• Two N/2-point DFTs
• 2(N/2)2 complex multiplications
• 2(N/2)2 complex additions
• Combining the DFT outputs
• N complex multiplications
• N complex additions
• Total complexity
• N2/2N complex multiplications
• N2/2N complex additions
• More efficient than direct DFT
• Repeat same process
• Divide N/2-point DFTs into
• Two N/4-point DFTs
• Combine outputs

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Decimation In Time Contd

• After two steps of decimation in time
• Repeat until were left with two-point DFTs

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Decimation-In-Time FFT Algorithm

• Final flow graph for 8-point decimation in time
• Complexity
• Nlog2N complex multiplications and additions

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Butterfly Computation

• Flow graph constitutes of butterflies
• We can implement each butterfly with one
  multiplication
• Final complexity for decimation-in-time FFT
• (N/2)log2N complex multiplications and additions

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In-Place Computation

• Decimation-in-time flow graphs require two sets
  of registers
• Input and output for each stage
• Note the arrangement of the input indices
• Bit reversed indexing

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Decimation-In-Frequency FFT Algorithm

• The DFT equation
• Split the DFT equation into even and odd
  frequency indexes
• Substitute variables to get
• Similarly for odd-numbered frequencies

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Decimation-In-Frequency FFT Algorithm

• Final flow graph for 8-point decimation in
  frequency

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Changing the Sampling Rate

• A continuous-time signal can be represented by
  its samples as
• We can use bandlimited interpolation to go back
  to the continuous-time signal from its samples
• Some applications require us to change the
  sampling rate
• Or to obtain a new discrete-time representation
  of the same continuous-time signal of the form
• The problem is to get xn given xn
• One way of accomplishing this is to
• Reconstruct the continuous-time signal from xn
• Resample the continuous-time signal using new
  rate to get xn
• This requires analog processing which is often
  undersired

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Sampling Rate Reduction by an Integer Factor
Downsampling

• We reduce the sampling rate of a sequence by
  sampling it
• This is accomplished with a sampling rate
  compressor
• We obtain xdn that is identical to what we
  would get by reconstructing the signal and
  resampling it with TMT
• There will be no aliasing if

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Frequency Domain Representation of Downsampling

• Recall the DTFT of xnxc(nT)
• The DTFT of the downsampled signal can similarly
  written as
• Lets represent the summation index as
• And finally

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Frequency Domain Representation of Downsampling
No Aliasing
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Frequency Domain Representation of Downsampling
w/ Prefilter
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Increasing the Sampling Rate by an Integer
Factor Upsampling

• We increase the sampling rate of a sequence
  interpolating it
• This is accomplished with a sampling rate
  expander
• We obtain xin that is identical to what we
  would get by reconstructing the signal and
  resampling it with TT/L
• Upsampling consists of two steps
• Expanding
• Interpolating

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Frequency Domain Representation of Expander

• The DTFT of xen can be written as
• The output of the expander is frequency-scaled

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Frequency Domain Representation of Interpolator

• The DTFT of the desired interpolated signals is
• The extrapolator output is given as
• To get interpolated signal we apply the following
  LPF

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Interpolator in Time Domain

• xin in a low-pass filtered version of xn
• The low-pass filter impulse response is
• Hence the interpolated signal is written as
• Note that
• Therefore the filter output can be written as

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Changing the Sampling Rate by Non-Integer Factor

• Combine decimation and interpolation for
  non-integer factors
• The two low-pass filters can be combined into a
  single one
• Application Filter Bank

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DSP board

• MEC 311
• Dec. 13th, 2007 Class
• You are welcomed to play after classes
• I will install the fixed point DPS there during
  Xmas break.
• Thanks for Dr. Welch, Barsha, Angus, and Lamont.