Ya Bao Fundamentals of Communication Theory Ya Bao Contact message: Room: T700B Telephone: Website: Assessment: 3-hour written examination – 70% Lab accessed report – 30% (by lab tutor)

Ya Bao Introduction of the unit This unit consists of five topics :  Signals and processes.  Fourier analysis and applications  Random signals and processes  Correlation processes.  Electrical noise

Ya Bao Chapter One. Signals and processes Learning outcomes You will be expected to know: the definitions of deterministic and non- deterministic signals; mathematical representations of deterministic signals; the idea of power and energy signal and methods to calculate these parameters; processes of multiplication, and convolution of time signals.

Ya Bao Introduction A Communication System

Ya Bao Sinusoidal expressions Sinusoidal signal Or, or Where: A is the sinusoid's amplitude ω is the angular velocity of the sinusoid in radian/s, θ is an arbitrary phase in radian. Note: ω=2πf

Ya Bao Time domain graph

Ya Bao Frequency domain spectra

Ya Bao Classification of signals Energy signals, Power signals An energy signal is a pulse-like signal that usually exits for only a finite interval of time or has a major portion of its energy concentrated in a finite time interval.

Ya Bao Classification of signals (cont1) An energy signal is defined to be one fro which the Is finite even when the time interval becomes infinite; i.e., when Average power dissipated by the signal f(t)

Ya Bao Classification of signals (cont2) Power signal Then the signal f(t) has finite average power and is called a power signal.

Ya Bao Classification of signals (cont3) Periodic, Nonperiodic (aperiodic) A periodic signal is one that repeats itself exactly after a fixed length of time. T – period, it define the duration of one complete cycle of f(t) If energy/cycle is finite then it is power signal. Any signal for which there is no value of period T is said nonperiodic (or aperiodic) signal.

Ya Bao Classification of signals (cont4) Deterministic, non-deterministic (random) Deterministic signal: no uncertainty in its values.  an explicit mathematical expression can be written Random signal: some degree of uncertainty before it actual occurs. (discussed later)  A collection of signals, each of which is different e.g. uncertain starting phase  Future values of the signal may not be predictable. E.g. noise

Ya Bao Multiplication and Convolution Multiplication in frequency domain is convolution in time domain. Convolution in frequency domain is multiplication in time domain. Convolution may be defined

Ya Bao Convolution (con1)-- example Example: Convolve the two signals g(t) and h(t) in (a)

Ya Bao Convolution (con2)– solution of example Step 1 Introduce a dummy variable to form g(u) and h(u) F(b) Step 2 Form g(t-u), F(c). Step 3 F(d). Place g(t-u) and h(u) on a common set of axes.

Ya Bao Convolution (con3)– solution of example Three distinct regimes:

Ya Bao Convolution (con4)– solution of example Step 4 Determine the convolution

Ya Bao Convolution (con5)– solution of example

Ya Bao Convolution (con6)– solution of example The results may be sketched to show pictorially the effect of convolving g(t) and h(t)

Ya Bao Properties of Convolution Commutative Law Distributive Law Associative Law

Ya Bao exercise Find the convolution of the rectangular pulse f 1 (t) and the triangular pulse f 2 (t) show in following Fig.

Ya Bao Analogue Filters Filter: a circuit that place a a limit upon the range of frequencies it will pass, and rejects any frequencies that fall outside this range. Low pass, high pass, band pass, band stop (notch) Commonly used in communication systems.

By Ya Bao

Ya Bao Bandwidth of a system Bandwidth W -- the interval of positive frequencies over which the magnitude H(w) remains within –3dB.