Information Theory

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Unit - III
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Fast Fourier Transform

• In 1967 calculation of a 8192-point DFT on the
  top-of-the line IBM 7094 took .
• 30 minutes using conventional techniques
• 5 seconds using FFTs
• Divide and conquer approach
• Radix-r algorithms

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Decimation-in-time algorithm

• Consider the FFT algorithm for an integer power
  of 2,
• Create separate sums for even and odd values of
  n
• Letting for n even and
  for n odd, we obtain

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Using following Properties of W-function
and
The DFT
F1(k) WNkF2 (k)
Have we gained anything? for N8 direct
computation will require (8)2 64
multiplications After decimation No of
multiplications (4)2 (4)2 8 40
X(k) F1(k) WNkF2 (k) and X(kN/2) F1(k) -
WNkF2 (k) (prove it)
X (r)
F1 (r)
F2 (r)
X (rN/2)
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Same butterfly is applicable for evaluating two
point DFT
Complete Butterfly diagram for 8-point FFT
X07
x0
x4
x2
x6
x1
x5
x3
x7
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Decimation in frequency
Divide x(n) from middle
Decimate in frequency
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8
Use of FFT in linear filtering and correlation

• An assignment

9
Quantization effects in the computation of DFT

• In hardware the register length is fixed to some
  value (b bits)
• Multiplication of two b bit data may produce
  2b bit data.
• The truncation/rounding off of 2b bit data to
  b bit data produces quantization errors.

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• Assuming that quantization error is uniformly
  distributed, the variance of error after a single
  multiplication will be
• s2e S2/12 S is step size.
• With b bits q2b levels can be represented thus
  step size (S) 1/q 2-b
• In Direct Computation, 4N real multiplications
  are required to evaluate one point of an N point
  DFT.
• The variance of quantization error s2q 4N s2e
• For N2v s2q 2-2(b-v/2)/3
• Every 4 fold increase in N requires one more bit
  to offset the additional quantization error.

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Quantization errors in FFT

• Each butterfly computation involves 1 complex
  i.e. 4 real multiplications, thus
• s2q 4Z s2e
• Where, Z The number of butterflies that affect
  the computation of any one value of DFT.
• In general Z N/2N/4N/8..21 2v-1
• Thus Z is approximately N
• FFT does not reduce the quantization error.

12
The scaling factor

• If X(k) gt1 it results in overflow (no register
  can store a magnitude greater than unity)
• Direct computation the upper bound is
• X(k)lt ?(0 to N-1)x(n) (As W cant be gt1)
• RHS lt1 if all samples are divided by N thus range
  of input (-1/N to 1/N) and S2/N
• Variance of Input signal sequence x(n)
• s2x (2/N)2/12 1/3N2
• Variance of O/P signal sequence X(k)
• s2X Ns2x 1/3N (each X(k) depends upon all N
  points of x(n))
• SQNR s2X/ s2q

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• In FFT, scaling can be done after each stage.
• Maximum O/P value from a butterfly lt twice the
  maximum I/P value
• Thus a scaling factor (½) is used after each
  stage.
• There are v steps thus signal will experience
  (1/2)v 1/N (same scaling)
• The scaling of quantization error from previous
  stage greatly improves the performance.
• It can be shown that
• s2q 2/3(2-2b)