CELLULAR COMMUNICATIONS
DSP Intro
Signals: quantization and sampling
Signals are everywhere
Encode speech signal (audio compression)
Transfer encode signals using RF signal (modulation)
Detect antenna signal Pack several calls into a single RF signal
from the antenna (multiple access) Improve faded signal (equalization) Adjust transmitted signal power to save
battery
What is signal?
Continuous signal Real valued-function of time x=x(t), t=0 is now,
t<0 is the past Can’t work with it in the computer But easy to analyze
Discrete signal A sequence s=s(n), n=0 is now Values are quantized (e.g. 256 possible values) Need a time scale: n=1 is 1ms, n=2 is 2 ms etc. Can process by computer (finite portion a time)
Discrete signal from continuous Sampling
Sample value of a continuous signal every fixed time interval
Quantization Represent the sampled value using fixed
number of levels (N=255)
Example:sampling
0 0sin(2 ) sin( )x f t w t 0 0sin(2 ) sin( )s f n t w n t
Example
Frequency Domain
*almost* any wave from sine waves
Frequency domain
Can decompose *almost* every signal into sum of sinusoids multiplied by a *weight*
Frequently domain=*weights* of sinusoids
Example: Upper case letter for frequency domain X(0)=0,X(1)=1,X(2)=0.4,X(3)=0 X is the spectrum of x
( ) 1 2 0 0( ) ( ) sin(2 ) 0.4*sin(2 2 )sum nx x n x n f n t f n t
Example: Sawtooth
Frequency Domain X(k)=1/k
Spectrum of sawtooth
Example: Box
X(n)=1/n (n is odd), X(n)=0 (n is even)
Spectrum of a linear combination Spectrum of x1+x2 is
Spectrum of x1+ Spectrum of x2
Frequency Domain
*Almost* every good periodic function can be represented by
Two series (numbers) describe the function Recall Taylor expansion (polynomial base) Discreet Fourier Transform takes function
and gives it’s Fourier representation Inverse DFT….
Representing Fourier Series
Coefficient of cosines and sinus
Cosine amplitude and phase Still two series, not convenient
,k ka b
,k ka
DFT summary
Can go back and forth from time-domain to frequency domain representation
Can be computed efficiently (FFT)
Signal Power in frequency and time domain (Parseval theorem)
Sampling theorem
Periodic Sampling
Discrete signals are obtained from continuous signals (acoustic/speech, RF) by sampling magnitude every fixed time period
How much should sampling period be for obtaining a good idea about the signal
Too much samples: need more CPU, power, clock etc.
Ambiguity problem
Ambiguity
Sample Frequency:
Digital sequence representing also represent infinitely many other sinusoids
1/s s sf t f
0 0 0( ) sin 2 sin 2 2 sin 2 ( )s s ss
kx n f nt f nt kn f nt
t
0( ) sin 2 ( )s sx n f kf nt
0f
0 sf kf
Aliasing
Suppose our signal is composed of sinusoids from 1kHz to 4KHz (with varying weights)
At sampling rate of 5 kHz we can discard 1kHz+5kHz and 4+5kHz as we know that signal has only up to 5kHz
At sampling rate of 2kHz we can distinguish between 1kHz and 3kHz which both are possible
Ambiguity in frequency domain
Nyquist sampling frequency
Signal band Avoid aliasing Nyquist sampling frequency Maximum frequency without aliasing
[ : ] [ : ]a b c h c hf f f f f f
a s b s b af f f f f f
2s hf f
2s
h
ff
Sampling low pass signals
A signal is within the known band of interest
But contains some noise with higher frequencies (above Nyquist frequency)
Spectrum of digital signal will be corrupted
Low Pass Filter
Time vs. Frequency Domain
Spectrum of the pulse
Time vs. Frequency
Short pulse in time domain->wide spectrum
Power Spectral Density(PSD)
2( ) ( )PSD f X f
PSD and Separation of signals
Discrete systems
Discrete System
Example: ( ) 2 ( ) 1y n x n
Operation with signals
Can add and subtract two signal Graphical representation
Summation
Linear Systems
Simple but powerful Easy to implement
Example
Example 1Hz+3Hz sine waves
Frequency domain vs. Time Domain Analyze a discrete system in time
domain What it does to the sequence x(n)
Analyze a discrete system in frequency domain What it does to the spectrum
Change in coefficient of various sinusoids of a signal
Example:1Hz+3Hz
Nonlinear Example: 1Hz+3Hz
f(x1+x2)!=f(x1)+f(x2)
Non-linear systems
Might introduce additional sinusoids not present in input
Results from interaction between input sinusoids
Difficult to analyze Sometimes are used in practice We stick to linear systems for a while
Time-Invariant Systems
Has no absolute clock
Example:
Example
Unit Time Delay
Time-Delay
Feasible system can’t look into a future at n=0 can’t produce x’(0)=y(4) only at n=4, can output x’(0)=y(4)
LTI: Linear Time Invariant
LTI is easy to analyze and build. Will focus on them
Analyzing LTI systems
LTI systems
Linear Time-Invariant Recall linear algebra
A vector space has basis vectors Linear operator completely defined by its
behavior on basis vectors
LTI need to specify only on a single basis vector
( ) ( )n n n k n kS x y S x y
( ) ( ) ( )n n n nS ax by aS x bS y
Vector Space of Signals
Shifted Unit Impulse(SUI) signal
Basis for representation of the digital signals
1,( )
0, 0m
n mu n
n
SUI are a basis
Representation
( ) ( ) ( )mm
x n x m u n
Impulse response
For time invariants systems
For linear systems
0 0( ( )) ( ) ( ( )) ( )m mS u n h n S u n h n
( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( )m mm m m
y n S x n S x m u n x m h n x m h n m
Finite Impulse Response
Filter
Impulse response
( ) ( 2) ( 1) ( )y n x n x n x n
( 1) 0
(0) 1
(1) 1
(2) 1
(3) 0
h
h
h
h
h
Infinite Impulse Response
1( ) ( 1) ( 1)
2( 1) 0
(1) 1/ 2
(2) 1/ 4
(3) 1/ 8
( ) 1/ 2n
y n x n y n
h
h
h
h
h n
Convolution with Finite Impulse
( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( )m mm m m
y n S x n S x m u n x m h n x m h n m
( ) (0) ( ) (1) (1 ) ... ( ) (0)y n x h n x h n x n h
Change Index
( ) ( ) (0) ( 1) (1) ... ( ) ( )y n x n k h x n k h x n h k
0
( ) ( ) ( )m
y n x n m h m x h
LTI system
The output of the LTI system is the result of the convolution between the input and the impulse response
Convolution0
( ) ( ) ( )m
c n x n m h m x h
Convolution in Frequency Domain x(t), y(t) are signals X(f), Y(f) are their spectrum What is the spectrum C(f) of Convolution theorem C=X*Y
(multiplication)
Convolution in the time domain===Multiplication in the frequency domain
c x y
What LTI does to a signal
Y=X*H Dump some sinusoids (|H(f)|<1) Boost other sinusoids (|H(f)|>1) Change phase of some sinusoids Never adds sinusoids that does not
existed in the input signal
y x h
Example: Moving average
Example: 3 points weighted
Example: simple avg,more points
Magic 16 points filter