Basics of Signal Processing
SIGNALSOURCE RECEIVER describe waves in terms of their significant features understand the way the waves originate effect of the waves will the people in the boat notice ? ACTION
frequency = 1/T speed of sound × T, where T is a period sine wave period (frequency) amplitude phase
sinecosine Phase
Sinusoidal grating of image
Fourier idea –describe the signal by a sum of other well defined signals TOTO
Fourier Series A periodic function as an infinite weighted sum of simpler periodic functions!
A good simple function T i =T 0 / i
e.t.c. ad infinitum
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 period T
Orthogonality
x x 0 ++ = = area is positive (T/2)area is zero
f(t) = sin(2πt/T) + 4sin(4πt/T) = T 0 T 1.5 = T 030
= 00b 1 T/200 …………… area=b 1 T/2 area=b 2 T/2
T=2 f(t) f(t) sin(2πt) f(t) sin(4πt) area = DCarea = b 1 T/2area = b 2 T/2
Spacing of spectral components is 1/T Periodicity in one domain (here time) implies discrete representation in the dual domain (here frequency) 01/T2/T frequency 01/T2/T frequency Phase spectrum Magnitude spectrum
Aperiodic signal Discrete spectrum becomes continuous (Fourier integral) 01/T 0 2/T 0 frequency 01/T 0 2/T 0 frequency Phase spectrum Magnitude spectrum Spacing of spectral components is f 0 =1/T 0
Magnitude spectrum of voiced speech signal frequency log | S( ) | F1F1 F2F2 F3F3 F4F4 f0f0
1/f 0 1/F 1 1/F 2 Waveform Logarithmic power spectrum f0f0 F1F1 F2F2 F3F3
PROCESSING One way of “signal processing”
Going digital
sampling 22 samples per 4.2 ms 0.19 ms per sample 5.26 kHz t s =1/f s
Sampling > 2 samples per period, f s > 2 f T = 10 ms (f = 1/T=100 Hz) Sinusoid is characterized by three parameters 1.Amplitude 2.Frequency 3.Phase We need at least three samples per the period
T = 10 ms (f = 1/T=100 Hz) t s = 7.5 ms (f s =133 Hz < 2f ) Undersampling T’ = 40 ms (f’= 25 Hz)
Sampling at the Nyquist frequency 2 samples per period, f s = 2 f Nyquist rate t s = 5 ms (f s =200 Hz) ? ?? f s > 2 f
Sampling of more complex signals period highest frequency component Sampling must be at the frequency which is higher than the twice the highest frequency component in the signal !!! f s > 2 f max
Sampling 1.Make sure you know what is the highest frequency in the signal spectrum f MAX 2.Chose sampling frequency f s > 2 f MAX NO NEED TO SAMPLE ANY FASTER !
Periodicity in one domain implies discrete representation in the dual domain 01/T2/T frequency Magnitude spectrum T
frequency F =1/t s f s = 1/T time tsts T Sampling in time implies periodicity in frequency ! Discrete and periodic in both domains (time and frequency) DISCRETE FOURIER TRANSFORM
Recovery of analog signal Digital-to-analog converter (“sample-and-hold”) Low-pass filtering
Quantization 11 levels 21 levels 111 levels
a part of vowel /a/ 16 levels (4 bits)32 levels (5 bits)4096 levels (12 bits)
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”