the discrete fourier transform. the fourier transform “the fourier transform is a mathematical...
TRANSCRIPT
The Discrete Fourier Transform
The Fourier Transform
• “The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum.”
– from http://en.wikipedia.org/wiki/Fourier_transform
The Fourier Transform
• “For instance, the transform of a musical chord made up of pure notes (without overtones) expressed as amplitude as a function of time, is a mathematical representation of the amplitudes and phases of the individual notes that make it up.”
– from http://en.wikipedia.org/wiki/Fourier_transform
Amplitude & phase
• f(x) = sin( x + ) +
– is the amplitude– is the frequency– is the phase– is the DC offset
More generally
• f(x) = 1 sin( 1 x + 1 ) + 2 sin( 2 x + 2 ) +
The Fourier Transform
• “The function of time is often called the time domain representation, and the frequency spectrum the frequency domain representation.”
– from http://en.wikipedia.org/wiki/Fourier_transform
Applications
• differential equations• geology• image and signal processing• optics• quantum mechanics• spectroscopy
REVIEW OF COMPLEX NUMBERS
Complex numbers
Complex numbers . . .
•extend the 1D number line to the 2D plane
•are numbers that can be put into the rectangular form, a+bi where i2 = -1, and a and b are real numbers.
Complex numbers(rectangular form)
yixz
Complex numbers
Complex numbers . . .•a is the real part; b is the imaginary part
•If a is 0, then a+bi is purely imaginary; if b is 0, then a+bi is a real number.
•originally called “fictitious” by Girolamo Cardano in the 16th century
Complex arithmetic
• add/subtract
– add/subtract the real and imaginary parts separately
Complex arithmetic
• complex conjugate
– often denoted as
– negate only the imaginary part
zor *z
Complex arithmetic
• inverse
wherez is a complex numberz bar is the length or magnitude of za is the real partb is the imaginary part
22
1
ba
z
zz
z
z
Complex arithmetic
• multiplication (FOIL)
iadbcbdacdicbia
Complex arithmetic
• division
dic
dic
dic
bia
idc
adbc
dc
bdac
dic
bia
2222
complex conjugate of denominator
Complex numbers(polar form)
sin
cos
andinteger an is
tanarg
where
sin cos
1
22
)2(
ry
rx
n
x
yz
yxzr
er
rir
er
iyxz
ni
i
exponential vs. trigonometric
2cos
2sin
sincos
ixix
ixix
ix
eex
i
eex
xixe
Leonhard Euler 1707-1783
(phasor form)
DFT(DISCRETE FOURIER TRANSFORM)
DFT
• Say we have a sequence of N (possibly complex) numbers, x0 … xN-1.
• The DFT produces a sequence of N (typically complex) numbers, X0 … XN-1, via the following:
1
0
2N
n
nink
Nk
exX
DFT & IDFT
• The DFT (Discrete Fourier Transform) produces a sequence of N (typically complex) numbers, X0 … XN-1, via the following:
• The IDFT (Inverse DFT) is defined as follows:
1
0
2N
n
nink
Nk
exX
1
0
21 N
k
nikk
Nk
eXN
x
Calculating the DFT
So how can we actually calculate ?
1
0
2N
n
nink
Nk
exX
Calculating the DFT
• So how can we calculate ?
• Let’s use this relationship:
• Then
• So what does this mean?
1
0
2N
n
nink
Nk
exX
1
0
2sin2cosN
nNk
Nk
nk ninxX
xixeix sincos
Interpretation of DFT
Back to the polar form:
– r/N is the amplitude and is the phase of a sinusoid with frequency k/N into which xn is decomposed
1
0
2sin2cosN
nNk
nNk
nk nixnxX
CALCULATING THE DFT USING EXCEL
x_n X_ 1 X_ 2N x b sin(b x) b2 sin(b2 x) 0.5*sin(b x) + 0.5*sin(b2 x) real imag real imag real imag
500 0 0.05 0.00 0.1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.001 0.05 0.10 0.07 0.07 0.00 0.07 0.00 0.07 0.002 0.10 0.20 0.15 0.15 0.00 0.15 0.00 0.15 -0.013 0.15 0.30 0.22 0.22 0.00 0.22 -0.01 0.22 -0.024 0.20 0.39 0.29 0.29 0.00 0.29 -0.01 0.29 -0.035 0.25 0.48 0.36 0.36 0.00 0.36 -0.02 0.36 -0.056 0.30 0.56 0.43 0.43 0.00 0.43 -0.03 0.43 -0.067 0.34 0.64 0.49 0.49 0.00 0.49 -0.04 0.49 -0.098 0.39 0.72 0.55 0.55 0.00 0.55 -0.06 0.54 -0.119 0.43 0.78 0.61 0.61 0.00 0.61 -0.07 0.59 -0.14
10 0.48 0.84 0.66 0.66 0.00 0.66 -0.08 0.64 -0.1611 0.52 0.89 0.71 0.71 0.00 0.70 -0.10 0.68 -0.1912 0.56 0.93 0.75 0.75 0.00 0.74 -0.11 0.71 -0.2213 0.61 0.96 0.78 0.78 0.00 0.77 -0.13 0.74 -0.2514 0.64 0.99 0.81 0.81 0.00 0.80 -0.14 0.76 -0.2815 0.68 1.00 0.84 0.84 0.00 0.82 -0.16 0.78 -0.3116 0.72 1.00 0.86 0.86 0.00 0.84 -0.17 0.79 -0.3417 0.75 0.99 0.87 0.87 0.00 0.85 -0.18 0.79 -0.3618 0.78 0.97 0.88 0.88 0.00 0.86 -0.20 0.79 -0.3819 0.81 0.95 0.88 0.88 0.00 0.85 -0.21 0.78 -0.4020 0.84 0.91 0.88 0.88 0.00 0.85 -0.22 0.77 -0.4221 0.87 0.86 0.87 0.87 0.00 0.84 -0.23 0.75 -0.4422 0.89 0.81 0.85 0.85 0.00 0.82 -0.23 0.72 -0.4523 0.91 0.75 0.83 0.83 0.00 0.79 -0.24 0.69 -0.4524 0.93 0.68 0.80 0.80 0.00 0.77 -0.24 0.66 -0.4625 0.95 0.60 0.77 0.77 0.00 0.74 -0.24 0.63 -0.4526 0.96 0.52 0.74 0.74 0.00 0.70 -0.24 0.59 -0.4527 0.98 0.43 0.70 0.70 0.00 0.66 -0.23 0.55 -0.4428 0.99 0.33 0.66 0.66 0.00 0.62 -0.23 0.50 -0.4329 0.99 0.24 0.62 0.62 0.00 0.58 -0.22 0.46 -0.4130 1.00 0.14 0.57 0.57 0.00 0.53 -0.21 0.42 -0.3931 1.00 0.04 0.52 0.52 0.00 0.48 -0.20 0.37 -0.37
X_0
Check w/ matlab/octave% see http://www.mathworks.com/help/matlab/ref/fft.html
N = 256; % # of samplesn = (0:N-1); % subscriptsb1 = 0.5; % freq 1b2 = 2.5; % freq 2
xn = 0.5 * sin( b1*n ) + 0.2 * sin( b2*n );
plot( xn );Xn = fft( xn );plot( abs(Xn(1:N/2)) );
X0real = xn .* cos( -2*pi*n*0/N );X0imag = xn .* sin ( -2*pi*n*0/N );
X1real = xn .* cos( -2*pi*n*1/N );X1imag = xn .* sin ( -2*pi*n*1/N );
X2real = xn .* cos( -2*pi*n*2/N );X2imag = xn .* sin ( -2*pi*n*2/N );
X3real = xn .* cos( -2*pi*n*3/N );X3imag = xn .* sin ( -2*pi*n*3/N );... Note: .* is element-wise (rather
than matrix) multiplication in matlab.
Add random noise.% see http://www.mathworks.com/help/matlab/ref/fft.html
N = 256; % # of samplesn = (0:N-1); % subscriptsb1 = 0.5; % freq 1b2 = 2.5; % freq 2r = randn( 1, N ); % noisexn = 0.5 * sin( b1*n ) + 0.2 * sin( b2*n ) + 0.5 * r;plot( xn );Xn = fft( xn );plot( abs(Xn(1:N/2)) );
X0real = xn .* cos( -2*pi*n*0/N );X0imag = xn .* sin ( -2*pi*n*0/N );
X1real = xn .* cos( -2*pi*n*1/N );X1imag = xn .* sin ( -2*pi*n*1/N );
X2real = xn .* cos( -2*pi*n*2/N );X2imag = xn .* sin ( -2*pi*n*2/N );
X3real = xn .* cos( -2*pi*n*3/N );X3imag = xn .* sin ( -2*pi*n*3/N );...
Signal without and with noise.
Signal with noise.FFT of noisy signal (two major components are still apparent).
Example of differences in phase.
xn = 0.5 * sin( b1*n ) + 0.2 * sin( b2*n )xn = 0.5 * sin( b1*n – 0.5 ) + 0.2 * sin( b2*n )
Computational complexity:DFT vs. FFT
• The DFT is O(N2) complex multiplications.• In 1965, Cooley (IBM) and Tukey (Princeton) described
the FFT, a fast way (O(N log2 N)) to compute the FT using digital computers.– It was later discovered that Gauss described this algorithm
in 1805, and others had “discovered” it as well before Cooley and Tukey.
– “With N = 106, for example, it is the difference between, roughly, 30 seconds of CPU time and 2 weeks of CPU time on a microsecond cycle time computer.” – from Numerical Recipes in C
Extending the DFT to 2D(and higher)
• Let f(x,y) be a 2D set of sampled points. Then the DFT of f is the following:
• (Note that engineers often use i for amps (current) so they use j for -1 instead.)
1
0
1
0
//2
1
0
1
0
//2
,,
,1
,
M
u
N
v
NvyMuxj
M
x
N
y
NvyMuxj
evuFyxf
eyxfMN
vuF
Extending the DFT to 2D(and higher)
• In fact, the 2D DFT is separable so it can be decomposed into a sequence of 1D DFTs.
• And this can be generalized to higher and higher dimensions as well.
1
0
/21
0
/2 ,11
,N
y
NvyjM
x
Nuxj eyxfN
eM
vuF
The classical “Gibbs phenomenon”
• Visit http://en.wikipedia.org/wiki/Square_wave.
• Hear it at http://www.youtube.com/watch?v=uIuJTWS2uvY.