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 

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

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

Complex Representation  Goal: single series. Trick: complex numbers  Euler identity  Negative frequency  Complex conjugate:  Two complex coefficients  Complex coefficients represent real signal

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

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

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)

PSD and Separation of signals

Discrete systems

Discrete System  Example:

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

Vector Space of Signals  Shifted Unit Impulse(SUI) signal  Basis for representation of the digital signals

SUI are a basis

Representation

Impulse response  For time invariants systems  For linear systems

Finite Impulse Response  Filter  Impulse response

Infinite Impulse Response

Convolution with Finite Impulse  Change Index

LTI system  The output of the LTI system is the result of the convolution between the input and the impulse response

Convolution

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

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

Example: Moving average

Example: 3 points weighted

Example: simple avg,more points

Magic 16 points filter