Real-Time Digital Signal Processing Laboratory Course Overview

Logo Description automatically generated Review for Midterm #2 Prof. Brian L. Evans EE 445S Real-Time Digital Signal Processing Laboratory 19 March 2025

2Outline Course objectives Design flow Signal processing Communication systems Communication system tradeoffs Symbol timing recovery

Course Overview 3 Objectives Build intuition for signal processing concepts Explore design tradeoffs in signal quality vs. run-time implementation complexity 6.6 6.5 6.4 6.3 6.2 6.1 6.0 5.9 5.8 5.7 6.6 6.5 Asymmetric Digital Subscriber Line (ADSL) Receiver Design FIR channel equalizer gives up to 10x increase in bit rate vs. not having one ADSL transmitter sends training data Eight adaptive channel equalizer design methods shown on right Three have best tradeoffs in bit rate vs. run-time computational complexity 6.4 6.3 SymMinSSNR SymMMSE 6.2 MinSSNR SymMinISI 6.1 MMSE MinISI 6 MDS DualPath 5.9 5.8 5.7 105 106 107 5.6 4.5 5 5.5 6 6.5 7 Receiver design: bit rate in Mbps vs. multiplications for eight channel equalizer training methods [Data from Figs. 6 & 7 in B. L. Evans et al., Unification and Evaluation of Equalization Structures , IEEE Trans. Sig. Proc., 2005]

Signal Quality Measures 4 Signal-to-noise ratio: SNR =Signal Power Noise Power Thermal noise (modeled as Gaussian) in Rx analog/RF front end Quantization noise (modeled as uniform) in data converters Communication Systems PAM/QAM Bit rate: ? ???? Nearest Tx Constellation Point or Training Symbol PAM Bit error rate ? ?0 ??? PAM Error vector magnitude squared ( ?? ??)2 Estimated Rx symbol amplitude Data Converters Signal-to-noise ratio Total harmonic distortion plus noise (lecture slide 8-14)

Design Flow 5 Matrices Speech & audio Fourier series Image & video Difference & differential equations Communications Fourier transforms Sampling Filtering Probability Modulation Adaptation Feedback run-time complexity analysis Builds on courses in programming, embedded systems, and signals & systems Feedback run-time complexity analysis

Signal Processing Building Blocks 6 Signals Impulse Sinusoids Exponentials Rectangular Pulse Triangular Pulse Sinc & Raised Cosine Chirp Pseudo-noise Impulse train Noise Systems Adder & gain/multiplier Ideal delay FIR & IIR filters Pointwise nonlinearities (squarer, absolute value, etc.) Signal generation (sinusoidal) Samplers & up/downsampling Quantizers Modulators/demodulators Adaptation (steepest descent) Fast Fourier transform

Increasing Sampling Rate 7 1 1 4 1 Input to Upsampler by 4 g[m] 4 n Upsampling by L denoted as L Outputs input sample followed by L-1 zeros Increases sampling rate by factor of L 0 1 2 Output of Upsampler by 4 m 0 1 2 3 4 5 6 7 8 Finite impulse response (FIR) filter g[m] Fills in zero values generated by upsampler Bandwidth / L Every Lth sample of g[m] is zero Multiplies by zero L-1 out of every L times Output of FIR Filter m 0 1 2 3 4 5 6 7 8 1 4 FIR Represented as rate changing FIR block 1 input sample produces 4 output samples 7

Polyphase Filter Bank 8 Oversampling filter a.k.a. sampler + pulse shaper a.k.a. linear interpolator g0[n] s(Ln) 1 L g1[n] L 1 s(Ln+1) g[m] L Multiplies by zero (L-1)/L of the time gL-1[n] s(Ln+(L-1)) Filter bank (right) avoids multiplication by zero Split filter g[m] into L shorter polyphase filters operating at the lower sampling rate (no loss in output precision) Saves factor of L in multiplications and previous inputs stored and increases parallelism by factor of L 8

9Decreasing Sampling Rate 1 1 4 1 Input to Downsampler g[m] 4 m Finite impulse response (FIR) filter g[m] Lowpass filter to enforce sampling theorem Bandwidth is / L 0 1 2 3 4 5 6 7 8 Output of Downsampler n Downsampling by L denoted as L Inputs L samples Outputs first sample and discards L-1 samples Decreases sampling rate by factor of L 0 1 2 4 1 Represented as rate changing FIR block FIR 4 input samples produce 1 output sample

Polyphase Filter Bank Form 10 1 M s(Ln) Undersampling filter a.k.a. Matched filter + sampling a.k.a. linear decimator L h0[n] y[n] s(Ln+1) 1 h1[n] h[m] L s[m] v[m] y[n] s[m] Outputs discarded (L-1)/L of the time hL-1[n] s(Ln+(L-1)) y[1] = v[L] = h[0] s[L] + h[1] s[L-1] + + h[L-1] s[1] + h[L] s[0] Filter bank only computes values output by downsampler Split filter h[m] into L shorter polyphase filters operating at the lower sampling rate (no loss in output precision) Reduces multiplications and increases parallelism by factor of L 10

Communication Systems 11 Message signal is information to be sent Information may be voice, music, images, video, data Low frequency (baseband) signal centered at DC Transmitter baseband processing includes lowpass filtering to enforce transmission band Transmitter analog/RF front end includes digital-to-analog converter, analog/RF upconverter, and transmit filter Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER 11

Communication Systems 12 Propagating signals experience attenuation & spreading w/ distance Model the environment Receiver analog/RF front end includes receive filter, carrier recovery, analog/RF downconverter, automatic gain control and analog-to-digital converter Receiver baseband processing extracts/enhances baseband signal Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER 12

Quadrature Amplitude Modulation 13 Transmitter Baseband Processing i[n] L gT[m] Index Pulse shaper (FIR filter) Serial/ parallel converter cos( 0m) Map to 2-D constellation + J 1 Bits L gT[m] q[n] L samples per symbol (upsampling) sin( 0m) Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER 13 In-phase i[n] and quadrature q[n] symbol amplitudes each come from PAM constellation

Transmitter Analog/RF Front End 14 LNA Low-Noise Amplifier Mixer: Reduce circuit complexity by replacing multiplication by cosine with sampling device (single transistor) BPF selects replica of baseband spectrum at fc D/A LNA LPF BPF fs cos(2 p fct) Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER Wavelength = c / fc Antenna Length?

Communication Channel Modeling 15 Channel Impairments Each path has its own attenuation & delay Frequency distortion due to multiple paths Additive thermal noise & interference More in Lecture 12 Propagation Paths Two bounces Line of sight One bounce Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER

Receiver Analog/RF Front End 16 Mixer: Reduce circuit complexity by replacing multiplication by cosine with sampling device (single transistor) LPF selects replica of bandpass spectrum at baseband LNA Low-Noise Amplifier LNA BPF LPF A/D fs cos(2 p fct) Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER

17Baseband Equivalent Channel Baseband discrete-time channel model Combines transmitter carrier circuits, physical channel and receiver carrier circuits One model uses cascade of gain, FIR filter, and additive noise 0 a + FIR noise Fast simulation Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER

Quad. Amplitude Demodulation 18 Receiver Baseband Processing iest[n] hopt[m] L Matched filter (FIR filter) Parallel/ serial converter cos( 0m) Decision Device heq[m] 1 J Channel equalizer (FIR filter) Symbol Bits qest[n] hopt[m] L L samples per symbol (downsampling) sin( 0m) Received Message Message Baseband Processing Analog/RF Front End Transmission Medium Analog/RF Front End Baseband Processing s(t) r(t) TRANSMITTER CHANNEL RECEIVER 18

19QAM Signal Quality Assumptions Q 2 2 Each symbol is equally likely 3 3 Channel only consists of additive noise 1 1 2 2 I White Gaussian noise with zero mean and variance 2 in in-phase and quadrature components Total noise power of 2 2 2 2 1 1 3 3 2 2 16-QAM Carrier frequency and phase recovery Symbol timing recovery 9 d d = 2 ( ) 3 P e Q T Q T sym sym 4 Probability of symbol error d proportion is sym T al to SNR Constellation spacing of 2d Symbol duration of Tsym

Communication System Tradeoffs 20 What happens to signal quality and run-time implementation complexity if PAM system parameter in first column increases? Indicate increase, decrease, or blank to mean no effect Assume other parameters in first column are not changing Fall 2020 Midterm #2.4 Parameter Transmission Bandwidth Bit Rate Symbol Error Rate ? Tx Power Consumption Run-Time Complexity increase B bits in A/D output & D/A input 2d constellation spacing in Volts fsym symbol rate in Hz J bits/symbol ? decrease increase increase increase increase increase increase increase increase increase L samples/symbol decrease increase decrease increase ?? symbol periods in pulse shape

Symbol Timing Recovery Spring 2021 Midterm #2.1 21 Consider a baseband PAM system using a rect. pulse shape Matched filter impulse response is also a rectangular pulse when there is no noise k can be negative Receiver needs to find the symbol timing offset Develop adaptive method to update in nth symbol period

Symbol Timing Recovery Spring 2021 Midterm #2.1 22 Maximize power at received symbol amplitude ?(?????+ ?) =1 2?2(?) Objective function: ? ? ? ? ? = ? ? ? + ? ? ? ?(?) ?0(?) ?2 ???? ?(?) is Gaussian random signal with 0 mean and variance ?2 ???? is constant regardless of sampling time ? ?? ? ? ?????+ ? Average noise power ? ? + 1 = ? ? + ? ?=? ? ? ???(?) ? ? + 1 = ? ? + ? ? ?????+ ?[?] ? ?=?????+?[?]

Symbol Timing Recovery Spring 2021 Midterm #2.1 23 differentiator circuit A differentiator circuit can be as implemented as an RC circuit where the output is tapped across the resistor RL circuit where the output is tapped across the inductor For h(t) being a rectangular pulse, we simplify derivative of y(t) by using a linear time-invariant model of differentiation with impulse response v(t) per JSK Appendix G.2: ? ???(?) = ?(?) (?) ?(? ) = ? ? ? = ?(?) ? ? ???? ?(?) ?(?) = ?(?) ? ? ???? How to handle many symbols? Send bits 11 What is y2(t)? y(?) kTsym t Tsym 2Tsym 3Tsym

24Data Conversion Analog-to-Digital Digital-to-Analog Discrete to Continuous Conversion Analog Lowpass Filter Analog Lowpass Filter Quantizer B B Sample at rate of fs fs A/D and D/A lowpass filter fstop < fs fpass 0.9 fstop Astop = SNRdB Apass = 20 log10(1- ) where is quantizer step size: mmax is max quantizer voltage Quantize to B bits Quantization error = noise SNRdB C0 + 6.02 B Dynamic range SNR Signal Power = SNR 10 log 2mmax 2B-1 10 dB Noise Power = signal noise SNR 10 log 10 log P P 10 10 dB = signal dB P noise dB P SNR dB