Lecture 3 Outline Announcements: No class Wednesday Friday lecture (1/17) start at 12:50pm Review of Last Lecture Communication System Block Diagram Performance Metrics Fundamental Rate Limits and Shannon Capacity Periodic Signals and Fourier Series

Review of Last Lecture Analog, digital, and binary signals Analog communication systems Convert analog information signals to modulated analog signals Digital communication systems Convert bits to modulated digital signals Communication system block diagram

Communication System Block Diagram Source encoder converts message into message signal or bits. Transmitter converts message signal or bits into format appropriate for channel transmission (analog/digital signal). Channel introduces distortion, noise, and interference. Receiver decodes received signal back to message signal. Source decoder decodes message signal back into original message. Source Decoder ChannelReceiver Transmitter Text Images Video Source Encoder

Performance Metrics Analog Communication Systems Metric is fidelity Want m(t)  m(t) Digital Communication Systems Metrics are data rate (R bps) and probability of bit error (P b =p(b  b)) Without noise, never make bit errors With noise, P b depends on signal and noise power, data rate, and channel characteristics. ^ ^

Data Rate Limits Data rate R limited by signal power, noise power, distortion, and bit error probability Without distortion or noise, can have infinite data rate with P b =0. Shannon capacity defines maximum possible data rate for systems with noise and distortion Rate achieved with bit error probability close to zero In white Gaussian noise channels, C=B log(1+P s /P N ) Does not show how to design real systems Shannon obtained C=32 Kbps for phone channels Get 1.5 Mbps with DSL by using more bandwidth

Periodic Signals x p (t) periodic if exists T such that x p (t)=x p (t+T) for all t. Smallest such T is fundamental period T 0 Any integer multiple of T 0 is a period of x p (t) Fundamental period defined as f 0 =1/T 0 Aperiodic signals are not periodic 0T0T0 2T 0 -T 0

Projection of Signals Fourier series representation Project periodic signals onto basis functions Periodic signal is weighted sum of these functions 0 T0T0 0 T0T0 0 T0T0 0T0T0 2T 0 -T 0 c1c1 c2c2 c3c3

Exponential Basis Functions Fourier series uses exponential basis fcns Fourier series representation The {c n }s are the Fourier Series coefficients These represent the frequency components of the periodic signal.

Main Points Transmitter converts information into a signal appropriate for transmission, receiver does reverse. Performance metric for analog systems is fidelity, for digital it is rate and error probability. Data rates over channels with noise have a fundamental capacity limit. Fourier series represents periodic signals as a weighted sum of exponential functions. The Fourier series coefficients are the frequency components of the periodic signal.