September 9, 2004 EE 615 Lecture 2 Review of Stochastic Processes Random Variables DSP, Digital Comm Dr. Uf Tureli Department of Electrical and Computer Engineering Stevens Institute of Technology Hoboken NJ 07030

September 9, 2004 Stochastic Processes Fundamentals Random Variables A mapping between a discrete or a random event and a real number. (not a variable!) Ensemble Average Average or Expected value behavior of a random variable.

September 9, 2004 Continous Random Variables Distribution function F X (a) of RV X is: Probability density function f X (a) f X (a) > 0

September 9, 2004 Discrete Random Variables and Probability  Random variable X assumes a value as a function from outcomes of a process which can not be determined in advance.  Sample space S of a random variable is the set of all possible values of the variable X.   : set of all outcomes and divide it into elementary events, or states

September 9, 2004 Expectation, Variance and Deviation The moments of a random variable define important characteristics of random variables: The first moment is the expectation E[X]= : Note: The expectation has a misleading name and is not always the value we expect to see most. In the case of the number on a dice the expectation is 3.5 which is not a value we will see at all!. The expectation is as a weighted average. The variance is defined by Var[x] = - 2 = M 2 - M 1 2. The standard deviation  = Var[x] 1/2 evaluates the “spread factor”or x in relation to the mean.

September 9, 2004 Ensemble Average Mean: Continuous Discrete Variance

September 9, 2004 Correlation & Covariance Crosscorrelation Covariance If or equal zero, correlation equals covariance

September 9, 2004 Random Process X, Y need not be separate events X,Y can be samples of process observed at different instants t_1, t_2

September 9, 2004 Independence vs. Uncorrelatedness R.V.s X, Y independent if Uncorrelated (Weaker condition), when R.V. X, Y uncorrelated if covariance is zero. Independent R.V. always uncorrelated. Uncorrelated R.V. may not be independent!

September 9, 2004 Random Processes Random process is a rule for assigning every outcome of a probabilistic event to a function Random process is an indexed sequence of R.V.s R.P. is stationary in strict sense, if all statistics are time invariant Wide Sense stationary if first and second order statistics are time invariant.

September 9, 2004 WSS Process Properties =constant, For Gaussian process, WSS implies strict stationarity For WSS:

September 9, 2004 MOdulation/DEModulation Modulation: Converting digital data into an analog signal. Demodulation: Converting an analog signal into digital data

September 9, 2004 DIGITAL SIGNAL DISCRETE WAVEFORM TWO DISCRETE STATES: 1-BIT & 0-BIT ON / OFF PULSE DATA COMMUNICATION USES MODEM TO TRANSLATE ANALOG TO DIGITAL, DIGITAL TO ANALOG

September 9, 2004 Digital Comm over Fading Channels Comm Theory 609: Design/ Performance of Digital Comm. In Additive White Gaussian noise New: Linear Filter Channel with AWGN Traditional Soln: Equalization Question: How should signals be designed for complex channels?

September 9, 2004 Statistical Characterization of Channels Digital Comm. Proakis, 4 th Edition, Ch.14 pp.800 Notice channel has time varying impulse response!

September 9, 2004 Propagation Models Channel model provides reliable base in system design and research For simulations and design, simple model preferred In transmitter/receiver (transceiver) design, not accurate but typical and worst case models most relevant

September 9, 2004 Major Channel Effects Propagation Loss is attenuation, also called path loss Time Dispersion: multiple reflections due to obstacles leading to multipaths Doppler Effects: Time variant nature due to mobility of objects in an environment

September 9, 2004 Propogation Loss Free space propagation: Loss (dB):S(d)=S_0+10a log_10 (d)+b, where a and b depend on operating frequency, environment, obstructed or direct line of sight, around 5 GHz, a=3.75, b=-6.5, such that for distances 10-50m, S= dB!

September 9, 2004 Noise White Noise Interference: Narrowband Interference Microwave EmissionFrequency Hopping

September 9, 2004 Multipath Propagation Natural obstacles, buildings, furnitures, etc. Each path:delay, attenuation, phase shift

September 9, 2004 Terminology Static Channel Impulse response k:path index, a_k:path gain, theta_k:path phase shift, tau_k:path delay

September 9, 2004 Power Delay Profile (PDP) PDP RMS Delay Spread

September 9, 2004 Coherence Bandwidth Autocorrelation of channel frequency response For class of channels with exponential delay profile, autocorrelation can be computed as a statistical expectation Coherence BW:

September 9, 2004 Flat vs Frequency Selective Fading For channel with exponential delay spread If BW > B_coh: Frequency selective fading If BW < B_coh: Flat fading

September 9, 2004 Effect of channel Transmitted signal f c :carrier frequency, j=sqrt(-1) Received signal

September 9, 2004 Time Variant Channels Correlation: Coherence Time:

September 9, 2004 Doppler Spectra Doppler Spectrum: T_coh ~ 1/ f_d

September 9, 2004 Example: OFDM Modulation

September 9, 2004 Multicarrier Modulation DFT/FFT to generate subcarriers Real representation: Complex:

September 9, 2004 Demodulation

September 9, 2004 IFFT for modulation N point transform N^2 operations (complexity grows quadratically) NlogN complexity in the FFT/IFFT (slightly faster than linear) Radix-4 butterfly

September 9, 2004 FFT Implementation Decimation in Time

September 9, 2004 Multicarrier System -Wireless Complex Transmission

September 9, 2004 Wireline, Baseband Transmission

September 9, 2004 Decimation Decimation in Time, vs Frequency

September 9, 2004 Scalability-repetetive structure Partial FFT, if you use a subset of transmitted carriers

September 9, 2004 Cyclic Extension Transmission in frequency domain (FFT) DFT properties Signal and channel linearly convolved Prefix and postfix extension

September 9, 2004 Cyclic Prefix Make the convolution linear Filtering: Cylic Prefix and Removal makes linear convolution into Circular convolution

September 9, 2004 Time/ Frequency Domain - Processing Why not equalize in frequency domain? Stu Schwartz (Princeton) Hikmet Sari (France Telekom) (w/cylic prefix) Falconer (Carleton)