Basics of Signal Processing
frequency = 1/T
speed of sound × T, where T is a period
sine wave •period (frequency)•amplitude•phase
Fourier Series
A periodic function as an infinite weighted sum of simpler periodic functions!
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f (t) = wi
i= 0
∞
∑ f i(t)
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f (t) = ki sin(iω0
i=1
∞
∑ + ϕ n )
= [bi
i=1
∞
∑ sin(iω0) + ai cos(iω0)]
= Re ˆ c i ⋅e− jω0n
i= 0
∞
∑ , ˆ c − complex
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f (t) = ki sin(iω0
i=1
∞
∑ + ϕ n ) = [bi
i=1
∞
∑ sin(iω0) + ai cos(iω0)]
T=1/f
e.t.c……
T=1/f
e.t.c……
Fourier’s Idea
Describe complicated function as a weighted sum of simpler functions!-simpler functions are known-weights can be found
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Simpler functions - sines and cosines are orthogonal on period T, i.e.
f(mt)⋅ f(nt) 0
T
∫ dt = 0 for m ≠ n
period T period T
Orthogonality
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sinnωt ⋅cosmωtdt = 00
T
∫
cosnωt ⋅cosmωtdt = 0 for n ≠ m and T2
0
T
∫ for n = m
sinnωt ⋅sin mωtdt = 0 for n ≠ m and T2
0
T
∫ for n = m
f(t) = 1.5 + 3sin(2πt/T) + 4sin(4πt/T)
= + +0 T
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1
Tf(t)dt
0
T
∫ =1.50 T
1.5
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b1 =2
Tf
0
T
∫ (t)sin(2πt
T)dt = 3 = + +0 T
0 3 0
One example of a function
=
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f(t) = DC + a i cos(2πit
T) + b1 sin(
2πit
T)
⎡ ⎣ ⎢
⎤ ⎦ ⎥=
i=1
∞
∑ DC + a1 cos(2πt
T) + b1 sin(
2πt
T) + a2 cos(
4πt
T) + b2 sin(
4πt
T) + a3 cos(
6πt
T) + b3 sin(
6πt
T) + .........
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f(t)sin(2πt
T)dt
0
T
∫ = {DC0
T
∫ sin(2πt
T) + a1 cos(
2πt
T)sin(
2πt
T) + b1 sin(
2πt
T)sin(
2πt
T) + a2 cos(
4πt
T)sin(
2πt
T) + b2 sin(
4πt
T)sin(
2πt
T) + .........}dt
0 0 b1T/2 0 0 ……………
area=b1T/2
area=b2T/2
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f(t) = DC + f1(t) + f2(t) = DC + b1 • sinωt + b2 • sin2ωt
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sin2
0
T
∫ (t
T) dt =
T
2
T=2
+
-+ +
+ + + +
+ + + +
+ +- -
- - - -
f(t) f(t) sin(2πt) f(t) sin(4πt)
area = DC area = b1T/2 area = b2T/2
Spacing of spectral components is 1/T
Periodicity in one domain (here time) implies discrete representation in the dual domain (here frequency)
0 1/T 2/Tfrequency
0 1/T 2/Tfrequency
Phase spectrumMagnitude spectrum
Aperiodic signal
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T0 →∞ ⇒ frequency spacing f0 → 0
Discrete spectrum becomes continuous (Fourier integral)
0 1/T0 2/T0frequency0 1/T0 2/T0frequency
Phase spectrumMagnitude spectrum
Spacing of spectral components is f0 =1/T0
Sampling
> 2 samples per period,fs > 2 f
T = 10 ms (f = 1/T=100 Hz)
Sinusoid is characterized by three parameters1.Amplitude2.Frequency3.Phase
We need at least three samples per the period
Sampling at the Nyquist frequency
2 samples per period,fs = 2 f
Nyquist rate
ts = 5 ms (fs=200 Hz)
?
??
fs > 2 f
Sampling of more complex signals
period period
highest frequencycomponent
Sampling must be at the frequency which is higher than the twice the highest frequency component in the signal !!!
fs > 2 fmax
Sampling
1. Make sure you know what is the highest frequency in the signal spectrum fMAX
2. Chose sampling frequency fs > 2 fMAX
NO NEED TO SAMPLE ANY FASTER !
Periodicity in one domain implies discrete representation in the dual domain
0 1/T 2/Tfrequency
Magnitude spectrumT
frequency
F =1/ts
fs = 1/T
time
ts
T
Sampling in time implies periodicity in frequency !
1
0
21 )()(
N
n
N
knj
N ekXnx
1
0
21 )()(
N
n
N
knj
N enxkX
Discrete and periodic in both domains (time and frequency)
DISCRETE FOURIER TRANSFORM
0.0000000000000000.3090167420035500.5877848229325430.8090165264524070.9510561882928811.0000000000000000.9510570082965530.8090180861922140.5877869697305400.3090192657165440.000000000000000-0.309014218288380-0.587782676130406-0.809014966706903-0.951055368282511-1.000000000000000-0.951057828293529-0.809019645926324-0.587789116524398-0.309021789427363-0.000000000000000
Quantization
• Quantization error = difference between the real value of the analog signal at sampling instants and the value we preserve
• Less error less “quantization distortion”
Homework 2
• Create for your fun (and education) various periodic functions using Fourier principle (and listen to the created signals)
Homework 5
• A signal generator produced a triangular wave which was sampled as indicated below. Was the signal samples correctly? If not, what went wrong and how would you fix it?
time