EE354 : Communications System I Lecture 2: Fourier review Aliazam Abbasfar

Outline Signals Fourier Series Fourier Transform Fourier properties Linear systems

Signals in communication systems m(t) m[n] m(t) m[n] x(t) y(t) message Source encoder Transmitter Source decoder Channel Receiver Analog systems m(t) is a continuous function Digital systems m[n] is a discrete function m[n] takes limited values DC level, energy, power x(t) t x(t) T t

Fourier series Periodic signals with period T0 f0 = 1/T0 : fundamental frequency cn :Line(discrete) spectrum of the signal Parseval’s theorem :

Fourier Transform Continuous spectrum Real signals : X(-f) = X*(f) Even signals : X(f) is real Odd signals : X(f) is imaginary

Rectangular pulse Rect(t) : a pulse with unit amplitude and width Sinc(f) = sin(pf)/(pf) Band-limited and time-limited signals

Fourier Transform Properties Useful properties Linearity Time shift Time/Freq. scaling Modulation Convolution/multiplication Differentiation/integration Duality:  Parseval’s equation : Energy and energy spectral density

Special signals DC x(t) = 1  X(f) = d(f) Impulse x(t) = d(t)  X(f) = 1 Sign x(t) = sgn(t)  X(f) = 1/jpf Step x(t) = s(t)  X(f) = 1/j2pf+ 1/2d(f) Sinusoids x(t) = ej2pf0t  X(f) = d(f-f0) Periodic signals .

Fourier Transform and LTI systems An LTI system is defined by its impulse response, h(t) H(f) : frequency response of system x(t) = ej2pf0t  y(t) = H(f0) ej2pf0t Eigen-functions and Eigen-values of any LTI system

Reading Carlson Ch. 2 and 3.1 Proakis 2.1, 2.2