Basics of Signal Processing

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 One example of a function

= 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

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”

Homework 2 Create for your fun (and education) various periodic functions using Fourier principle (and listen to the created signals)

Homework 3 What happens if the signal is not periodic and why?

Homework 4 How would you transmit the signal below and why? t0t0 t0t0

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