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

f (t) Asin(2t /T )

f (t) Asin(t )

Asin(t /2)

Acos(t)

sine cosinePhase

Sinusoidal grating of image

• Fourier idea– describe the

signal by a sum of other well defined signals

TO

Fourier Series

A periodic function as an infinite weighted sum of simpler periodic functions!

f (t) wi

i0

f i(t)

A good simple function

Ti=T0 / i

f i(t ) sin(i0t ),

where 0 2 / T0

f (t) ki sin(i0

i1

n )

[bi

i1

sin(i0) ai cos(i0)]

Re ˆ c i e j0n

i0

, ˆ c complex

e.t.c. ad infinitum

f (t) ki sin(i0

i1

n ) [bi

i1

sin(i0) ai cos(i0)]

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

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

0+

-

0+

-

0+

-

x

0 + +--

x

0

+ +

=

0 + +- -

=

area is positive (T/2) area is zero

=

f(t) DC a i cos(2it

T) b1 sin(

2it

T)

i1

DCa1 cos(2t

T) b1 sin(

2t

T) a2 cos(

4t

T) b2 sin(

4t

T) a3 cos(

6t

T) b3 sin(

6t

T) .........

f(t)sin(2t

T)dt

0

T

{DC0

T

sin(2t

T) a1 cos(

2t

T)sin(

2t

T) b1 sin(

2t

T)sin(

2t

T) a2 cos(

4t

T)sin(

2t

T) b2 sin(

4t

T)sin(

2t

T) .........}dt

0 0 b1T/2 0 0 ……………

area=b1T/2

area=b2T/2

f(t) DC f1(t) f2(t) DC b1 sint b2 sin2t

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

Aperiodic signal

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

22 samples per 4.2 ms 0.19 ms per sample 5.26 kHz

ts=1/fs

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

T = 10 ms (f = 1/T=100 Hz)

ts = 7.5 ms (fs=133 Hz < 2f )

Undersampling

T’ = 40 ms(f’= 25 Hz)

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

Recovery of analog signalDigital-to-analog converter (“sample-and-hold”)

Low-pass filtering

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

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”