Life in the frequency domain
Jean Baptiste Joseph
Fourier (1768-1830)
Spectrogram, Northern Cardinal
Sampling
  frequency in radians / sec
f  frequency in cycles / sec
T  period  1 / f  2 /  sec/ cycle
Sampling
if Ts  sampling interval,
Ts  T / 2, or
fs  2 f
( Nyquist theorem )
Example: CD rate f = 44,100 Hz,
so the highest frequency that
can be represented is 22,050 Hz
Called Nyquist frequency =
½ cycle/sample = f/2 Hz
Sampling and aliasing
Aliasing a sine wave:
• One frequency can masquerade as
another
• When viewed as a strobed phasor, it’s
easy to see that we need to sample at
least twice each period to capture the
frequency unambiguously
• Nyquist’s theorem: Highest allowed
signal frequency is half the sampling
frequency = Nyquist frequency
Aliasing strikes!
• Prefiltering: avoids aliasing on A-to-D
• Oversampling: can substitute cheap
digital filtering for expensive analog
filtering for A-to-D or D-to-A
conversion
The DFT and FFT
Think of the n samples
of the signal as n
points on a circle
And also think of n
frequency points as n
points on the circle
The Discrete Fourier Transform
(DFT)
N 1
1
i k t
xt   X k e ,
N k 0
N 1
X k   xt e
i  k t
,
t  0,..., N  1
k  0,..., N  1
representation
transform
t 0
k 
k 2
N
,
k  0,..., N  1
Frequencies on circle
Johann Carl Friedrich Gauss (1777-1855)
Heideman, Johnson, Burrus (1985)
The FFT
Divide-and-conquer algorithm for DFT
T ( N )  2T ( N / 2)  cN
Yields O(N log N) algorithm
FFT Butterfly
A once-a-minute application: JPEG
Discrete
Cosine
Transform
(DCT)
Steven W.
Smith's
Book
(1997)