Life in the frequency domain

Jean Baptiste Joseph Fourier (1768-1830)

Spectrogram, Northern Cardinal

Sampling

cyclesec///1period

sec/cyclesinfrequency

sec/radiansinfrequency

2

fT

f

Sampling

)theoremNyquist(2

or,2/

interval,samplingif

ff

TT

T

s

s

s

Example: CD rate f = 44,100 Hz, so the highest frequency thatcan 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)

1,...,0,2

1,...,0,

1,...,0,

1

0

1

0

1

NkN

kk

Nk

Nt

tiN

ttk

tiN

kkt

k

k

exX

eXN

x

representation

transform

Frequencies on circle

Heideman, Johnson, Burrus (1985)

Johann Carl Friedrich Gauss (1777-1855)

The FFT

cNNTNT )2/(2)(

Divide-and-conquer algorithm for DFT

Yields O(N log N) algorithm

FFT Butterfly

A once-a-minute application: JPEG

Steven W. Smith's Book (1997)

Discrete Cosine Transform(DCT)