Fourier Transforms

1
Fourier Transforms

• Discrete Fourier Transform (DFT) Algorithms
• Fast Fourier Transform (FFT) Algorithms

2
Discrete Fourier Transform (DFT)

• Fourier Transform assumes a continuous signal of
  infinite duration and bandwidth.
• For practical representation, we must sample in
  time and frequency, and amplitude must be
  quantized.
• Discrete Fourier Transform assumes N samples in
  time and frequency.
• DFT Xk S xn ej2?kn/N S xn WNkn
• IDFT xn (1/N) S Xk e j2?kn/N (1/N) S
  Xk WN kn
• where WNkn e j2?kn/N

N-1
N-1
n0
n0
N-1
N-1
k0
k0
3
Effect of Sampling in Time and Frequency

• By sampling in time, we get a periodic spectrum
  with the sampling frequency fs. The approximation
  of a Fourier transform by a DFT is reasonable
  only if the frequency components of x(t) are
  concentrated on a smaller range than the Nyquist
  frequency fs/2.

4

• By sampling in the frequency domain, the time
  function becomes periodic, i.e., the DFT assumes
  the time series to be periodic. If an N-sample
  DFT is applied to a signal that does not complete
  an integer number of cycles with an N-sample
  window, a phenomenon called leakage occurs.

5

• If possible, we should choose the sampling
  frequency and the analysis window in such a way
  that it covers an integer number of periods of
  x(t), if x(t) is periodic.
• We can also decrease leakage by using a window
  function that tapers smoothly to zero on both
  sides. See Example 6.1 Windowing.

6
Vector/matrix notation and an alternative
inversion algorithm

• X Wx ? x (1/N) WX
• x (1/N) WX (1/N) (WX)
• When computing IDFT, we can compute the DFT of
  X, i.e., WX. The result of IDFT is the
  conjugate of the result of WX scaled by (1/N).

7
The Fast Fourier Transform (FFT) Algorithms

• FFT uses different multidimensional index maps of
  the input and output sequences.
• Two-dimensional index transform
• n An1 Bn2 mod N
• where 0 ? n1 ? N11, 0 ? n2 ? N21 and
• N N1 N2
• See equation (6.18) on page 259.

8
The Cooley-Tukey FFT Algorithm

• The Cooley-Tukey FFT is the most universal of all
  FFT algorithms, because any factorization of N is
  possible.

9

• Index transform (index mapping)
• n N2n1 n2
• k k1 N1k2
• where 0 ? n1 ? N11, 0 ? n2 ? N21,
• 0 ? k1 ? N11, 0 ? k2 ? N21 and
• N N1 N2
• Xk S xn ej2?kn/N S xn WNkn
• WNkn WN(k1 N1k2 )( N2n1 n2 )
• WNN2n1k1 N1N2n1k2 n2 k1 N1n2k2

N-1
N-1
n0
n0
10

• WNN2n1k1 ej2? N2n1k1 /N ej2? n1k1 /N1 WN
  1n1k1
• WN N1N2n1k2 ej2? N1N2n1k2 /N ej2? n1k2 1
• WN N1n2k2 ej2? N1n2k2 /N ej2? n2k2 /N2 WN
  2n2k2
• Xk1,k2 S WN 2n2k2 (WN n2k1 S xn1,n2 WN
  1n1k1 )
• S WN 2n2k2 xn2,k1

N2-1
N1-1
n10
n20
N2-1
n20
11
Algorithm 6.8 Cooley-Tukey Algorithm

• An N N1 N2 -point DFT can be done using the
  following steps
• Compute an index transform of the input sequence
  according to n N2n1 n2
• Compute the N2 DFTs of length N1.
• Apply the twiddle factors WN n2k1 to the output
  of the first transform stage.
• Compute the N1 DFTs of length N2.
• Compute an index transform of the output sequence
  according to k k1 N1k2
• See Example 6.9 Cooley-Tukey FFT for N 12.

12
Radix-r Cooley-Tukey Algorithm

• Radix-r algorithms, N rs , S is the number of
  stages.
• The most popular algorithms are of basis r 2 or
  r 4, because the necessary basic DFTs can be
  implemented without any multiplications.

13
2-point DFT, Butterfly

• X(0) x(0)e0 x(1)e0
• x(0) x(1)
• X(1) x(0)e0 x(1)ej2? /2
• x(0) ? x(1)

x(0)
X(0)
x(1)
X(1)
14
4-point FFT

• n 2n1 n2
• k k1 2k2

x(0)
X(0)
x(2)
X(2)
W0
x(1)
X(1)
W2
x(3)
X(3)
15
Rearrange the positions of x(1) and x(2)
x(0)
X(0)
x(1)
X(2)
?1
W0
x(2)
X(1)
W2
x(3)
X(3)
?1
Group
16
Decimation-in-frequency (DIF) FFT algorithm

• See Fig. 6.12 and Table 6.3.
• The Decimation-in-frequency algorithm starts in
  the frequency domain to split the original DFT
  into shorter DFTs.
• The input values typically occur in natural
  order, while the index of the frequency values is
  in bit-reversed order.
• The decimation-in-time (DIT) FFT algorithm can be
  found in the book Fundamentals of Signals and
  Systems using the WEB and MATLAB, pages 339-342.

17
Number of complex multiplications and
add/subtract operations

• For r 2 and S stages (S log2 N), the number
  of complex twiddle factor multiplications is
• (S 1)?N/2 ((log2 N) 1) ? N/2.
• The number of complex add/subtract operations is
  
• S?N (log2 N) ? N.
• Compare with the direct DFT computation
• Complex multiplications N?N
• Complex add/subtract operations N?(N 1)

18

• Index transform for radix-2 Cooley-Tukey FFT
  using bit reversing
• Reversing of the bit sequence of the index.
• Example
• For the 8-point FFT in Fig. 6.12, the output at
  the second position (index 0012) of the output is
  X4 (index 1002) after bit reversing.
• Index transform for radix-4 Cooley-Tukey FFT
  using digit reversing
• Build digits of two bitsand then reverse the
  order of these digits. See Fig. 6.13.

19
Radix-2 Cooley-Tukey Algorithm Implementation

• Butterfly processor
• Complex multiplication
• Complex multiplication with the twiddle factor is
  often implemented with 4 real multiplications and
  two add/subtract operations.
• A more efficient complex multiplier is
  implemented with 3 real multiplications and 3
  add/subtract operations.
• See Algorithm 6.10.