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EE445S Real-Time Digital Signal Processing Lab Spring 2011 Digital Pulse Amplitude Modulation (PAM) Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Lecture 14 Outline • Introduction • Pulse shaping • Pulse shaping filter bank • Design tradeoffs • Optional sections Symbol recovery Communication system examples 14 - 2 Introduction • Each data symbol contains J bits of information • Modulate one of M = 2J discrete messages onto amplitude of waveform in each symbol period Tsym Known as M-level PAM or M-PAM for short 01 3d Bit rate is J fsym , where symbol rate fsym = 1 / Tsym Uniformly spaced amplitudes 00 d M M ai d (2i 1) where i 1 , ...,0 , ..., 10 d 2 2 Multiple ways to map symbols to amplitudes 3 d 11 Pulse Serial/ Map to PAM Impulse 1 Parallel J constellation ai modulator shaper 4-PAM gT(t) s*(t) bit J bits per symbol sampling transmitted Constellation stream symbol amplitude rate waveform 14 - 3 Pulse Shaping • Without pulse shaping, one would modulate using an impulse train, which uses infinite bandwidth s (t ) * a k k (t k Tsym ) Non-overlapping impulses • Limit bandwidth by pulse shaping At each time t, index k is filter with impulse response gT(t) indexed over number of a gTsym t k Tsym pulses overlapping in s * (t ) k one symbol period k Summation has finite number of terms Related to discrete-time convolution L samples per symbol period • Sample: let t = n L + m n is symbol index s * Ln m a gTsym L (n k ) m m is sample index in symbol k k (i.e. m = 0, 1, …, L-1) 14 - 4 Pulse Shaping Example 2-PAM with Raised Cosine Pulse Shaping 14 - 5 Pulse Shaping Block Diagram an s*(t) Transmit L gTsym[m] D/A Filter symbol sampling sampling analog analog rate rate rate • Upsampling by L denoted as L Outputs input sample followed by L-1 zeros Upsampling by converts symbol rate to sampling rate • Pulse shaping (FIR) filter gTsym[m] Fills in zero values generated by upsampler Multiplies by zero most of time (L-1 out of every L times) 14 - 6 Digital Interpolation Example Input to Upsampler by 4 16 bits 16 bits 28 bits 4 FIR Filter 44.1 kHz 176.4 kHz 176.4 kHz n 0 1 2 Digital 4x Oversampling Filter Output of Upsampler by 4 • Upsampling by 4 (denoted by 4) n’ Output input sample followed by 3 zeros 0 1 2 3 4 5 6 7 8 Four times the samples on output as input Output of FIR Filter Increases sampling rate by factor of 4 n’ • FIR filter performs interpolation 0 1 2 3 4 5 6 7 8 Lowpass filter with stopband frequency wstopband p / 4 For fsampling = 176.4 kHz, w = p / 4 corresponds to 22.05 kHz 14 - 7 Pulse Shaping Filter Bank Example • L = 4 samples per symbol • Pulse shape g[m] lasts for 2 symbols (8 samples) bits …a2a1a0 …000a1000a0 encoding ↑4 g[m] x[m] s[m] s[m] = x[m] * g[m] s[0] = a0 g[0] s[4] = a0 g[4] + a1 g[0] s[1] = a0 g[1] s[5] = a0 g[5] + a1 g[1] s[2] = a0 g[2] s[6] = a0 g[6] + a1 g[2] s[3] = a0 g[3] s[7] = a0 g[7] + a1 g[3] L polyphase filters …,s[4],s[0] {g[0],g[4]} m=0 …,s[5],s[1] …,a1,a0 {g[1],g[5]} s[m] Filter …,s[6],s[2] {g[2],g[6]} Bank …,s[7],s[3] Commutator 14 - 8 {g[3],g[7]} (Periodic) Pulse Shaping Filter Bank an Transmit L gTsym[m] D/A Filter symbol sampling sampling analog analog rate rate rate • Simplify by avoiding multiplication by zero Split long pulse shaping filter into L short polyphase filters operating at symbol rate gTsym,0[n] s(Ln) Transmit gTsym,1[n] s(Ln+1) D/A Filter an Filter Bank gTsym,L-1[n] Implementation 14 - 9 s(Ln+(L-1)) Pulse Shaping Filter Bank Example • Pulse length 24 samples and L = 4 samples/symbol n 3 s * L n m a k g T L ( n k ) m Six pulses contribute k n2 to each output sample • Derivation in Tretter's manual, m n 3 m s * nTsym Tsym ak gT nTsym Tsym kTsym m 0, 1,...,L 1 L k n2 L • Define mth polyphase filter m gTsym ,m [n] gTsym nTsym Tsym m 0, 1,...,L 1 L • Four six-tap polyphase filters (next slide) n 3 s * nTsym Tsym ak gTsym ,m n k m L k n2 14 - 10 Pulse Shaping Filter Bank Example 24 samples gTsym,0[n] in pulse 4 samples per symbol Polyphase filter 0 response is the first sample of the pulse shape plus every fourth sample after that x marks samples of polyphase filter 14 - 11 Polyphase filter 0 has only one non-zero sample. Pulse Shaping Filter Bank Example 24 samples gTsym,1[n] in pulse 4 samples per symbol Polyphase filter 1 response is the second sample of the pulse shape plus every fourth sample after that x marks samples of polyphase filter 14 - 12 Pulse Shaping Filter Bank Example 24 samples gTsym,2[n] in pulse 4 samples per symbol Polyphase filter 2 response is the third sample of the pulse shape plus every fourth sample after that x marks samples of polyphase filter 14 - 13 Pulse Shaping Filter Bank Example 24 samples gTsym,3[n] in pulse 4 samples per symbol Polyphase filter 3 response is the fourth sample of the pulse shape plus every fourth sample after that x marks samples of polyphase filter 14 - 14 Pulse Shaping Design Tradeoffs Computation Memory Memory Memory in MACs/s size in reads in writes in words words/s words/s Direct structure (slide 14-6) Filter bank structure (slide 14-9) fsym symbol rate L samples/symbol Ng duration of pulse shape in symbol periods 14 - 15 Optional Symbol Clock Recovery • Transmitter and receiver normally have different crystal oscillators • Critical for receiver to sample at correct time instances to have max signal power and min ISI • Receiver should try to synchronize with transmitter clock (symbol frequency and phase) First extract clock information from received signal Then either adjust analog-to-digital converter or interpolate • Next slides develop adjustment to A/D converter • Also, see Handout M in the reader 14 - 16 Optional Symbol Clock Recovery • g1(t) is impulse response of LTI composite channel of pulse shaper, noise-free channel, receive filter q (t ) s (t ) g1 (t ) * a k k g1 (t kTsym ) s*(t) is transmitted signal g1(t) is p(t ) q 2 (t ) ak am g1 (t kTsym ) g1 (t mTsym ) k m deterministic E{ p(t )} E{a k m k am } g1 (t kTsym ) g1 (t mTsym ) E{ak am} = a2 [k-m] a 2 g (t kT k 1 2 sym ) Periodic with period Tsym p(t) Receive BPF x(t) Squarer PLL B(w) H(w) 14 - 17 q(t) q2(t) z(t) Optional Symbol Clock Recovery • Fourier series representation of E{ p(t) } j k w sym t 1 Tsym j k w sym t E{ p (t )} k pk e where pk Tsym 0 E{ p (t )}e dt • In terms of g1(t) and using Parseval’s relation G1 w G1 kw sym w dw a2 a2 g12 t e sym dt jkw t pk Tsym 2p Tsym • Fourier series representation of E{ z(t) } zk pk H kw sym H kw sym G1 w G1 kw sym w dw 2 a 2pTsym p(t) Receive BPF x(t) Squarer PLL B(w) H(w) 14 - 18 q(t) q2(t) z(t) Optional Symbol Clock Recovery • With G1(w) = X(w) B(w) Choose B(w) to pass ½wsym pk = 0 except k = -1, 0, 1 Z k pk H kw sym H kw sym G1 w G1 kw sym w dw a2 2p Tsym Choose H(w) to pass wsym Zk = 0 except k = -1, 1 Ezt Z k e jkw symt jw symt jw symt e e w 2 cos( symt ) k • B(w) is lowpass filter with wpassband = ½wsym • H(w) is bandpass filter with center frequency wsym p(t) Receive BPF x(t) Squarer PLL B(w) H(w) 14 - 19 q(t) q2(t) z(t)