Lecture 3 Outline

Announcements:No class WednesdayFriday 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 systemsConvert analog information signals to

modulated analog signals

Digital communication systemsConvert 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.

SourceDecoderChannel ReceiverTransmitter

TextImagesVideo

)(tx )(ˆ tx)(ˆ...ˆˆ

21

tmbb

)(...21

tmbb

SourceEncoder

Performance Metrics

Analog Communication SystemsMetric is fidelityWant m(t)m(t)

Digital Communication SystemsMetrics are data rate (R bps) and

probability of bit error (Pb=p(bb))Without noise, never make bit errorsWith noise, Pb 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 Pb=0.

Shannon capacity defines maximum possible data rate for systems with noise and distortionRate achieved with bit error probability close to

zero In white Gaussian noise channels, C=B log(1+Ps/PN)Does not show how to design real systems

Shannon obtained C=32 Kbps for phone channelsGet 1.5 Mbps with DSL by using more bandwidth

Periodic Signals

xp(t) periodic if exists T such that

xp(t)=xp(t+T) for all t.

Smallest such T is fundamental period T0

Any integer multiple of T0 is a period of xp(t) Fundamental period defined as f0=1/T0

Aperiodic signals are not periodic

0 T0 2T0-T0

Projection of Signals

Fourier series representationProject periodic signals onto basis

functionsPeriodic signal is weighted sum of

these functions

0 T00 T0

0 T0

0 T0 2T0-T0

c1

c2c3

Exponential Basis Functions

Fourier series uses exponential basis fcns

Fourier series representation

The {cn}s are the Fourier Series

coefficientsThese represent the frequency components

of the periodic signal.

0/2)( Tntj

nnp ectx

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.