Lecture 13,14: Modulation Bandpass signals Aliazam Abbasfar

Outline Modulation Bandpass signals Bandpass processes

Modulation Message signal m(t) modulates a carrier signal x c (t) Convert lowpass message to bandpass signal Sinusoid carrier : x c (t) = A c cos( w c t +  c ) A c : carrier amplitude f c / c : carrier frequency/phase AM/FM/PM ASK/FSK/PSK Pulse carrier : PAM/PWM/PPM Linear/Non-linear modulations

Why modulation ? Sending messages in passband channels Allocated spectrum Better channel characteristics Design convenience Transmission of several messages simultaneously Frequency division multiplexing (FDM)

Bandpass signals

Bandpass signals (2) Equivalent lowpass signal v I (t) and v Q (t) are real, lowpass signals

Hilbert transform One-sided spectrum Hilbert transform H(f) = -j sgn(f)  h(t) = 1/t Quadrature filter: 90 phase shifter Lowpass signal

Bandpass transmission Equivalent lowpass channel If X lp (f) is band limited Narrowband/Wideband systems (B/f c )

Modulation/Demodulation Transmitter (modulator) message signals are constructed as lowpass signals Modulators generate bandpass signals Receiver (demodulator) bandpass received signals are demodulated to produce lowpass signals Lowpass signals are processed to get messages Lowpass to bandpass Amplitude  Envelope Constant phase  Carrier phase Linear phase  Carrier frequency offset Delay  Envelope(group) delay Baseband transmission f c = 0 (No modulation) Lowpass signal = real

Bandpass process X(t) is bandpass if G X (f)= 0 for |f-f c |>W The modulated signal The filtered noise Generalize bandpass signals If X(t) is zero-mean stationary process, X I (t) and X Q (t) are zero-mean and jointly stationary G Xi (f)= G Xq (f)= G X (f-f c ) + G X (f+f c )|f|<f c = 0 |f|>f c X I (t 0 ) and X Q (t 0 ) are uncorrelated Envelope and phase processes

Bandpass WGN process n(t) = n I (t) cos(  c t) – n Q (t) sin(  c t) Bandwidth 2W n I (t) and n Q (t) are independent and jointly Gaussian A(t) : Rayleigh distributed (t) : uniform distributed If f c is in the middle of the band G ni (f)= G nq (f)= N 0 |f|<W n I and n Q are independent If f c is on either end of the band G ni (f)= G nq (f)= N 0 /2|f|<2W P ni = P nq = P n = 2 N 0 W (n I +j n Q )= CN(0,4N 0 W)

Reading Carlson Ch. 4.1 and 3.6 Proakis 2.5, 3.1, 3.2