1. Wireless Networking and Communications Group
Reducing Complexity in Signal Processing Algorithms for Communication Receiver and Image Display Software
Brian L. Evans
Prof. Brian L. Evans
27 July 2010
Seminar at the American University of Beirut

2. Outline 2004 2005 2006 2007 2008 2009 2010 Embedded digital systems
Generating sinusoidal waveforms
Discrete-time filters
Multicarrier equalizers
Image halftoning algorithms
Conclusion
2004
2005
2006
2007
2008
2009
2010

3. Embedded Digital Systems
Often work on application-specific tasks
In consumer products (2008 units)
1200M cell phones M DSL modems
300M PCs M cars/light trucks
100M digital cameras M gaming consoles (2007)
100M DVD players
iPhone has six programmable processors (2008)
Embedded programmable processors
Inexpensive with small area and volume
Predictable off-chip input/output (I/O) rates
“Low” power (TI C MHz)
Limited on- chip memory
Fixed-point arithmetic

4. Embedded Digital Systems
Memory access in processors
External I/O: block data transfers to/from on-chip memory
Internal I/O: on-chip memory to CPU registers using data buses (e.g. TI C6000 processor has two 32-bit data buses)
Common word sizes for signal processing software
64-bit floating-point for desktop computing (e.g. Matlab)
32-bit floating-point for pro-audio and sonar beamforming
16-bit fixed-point for speech, consumer audio, image proc.
IEEE floating-point operations
Handles many special cases (e.g. +∞, -∞ and not a number)
Add, multiply, divide have comparable hardware complexity

5. Embedded Digital Systems
Fixed-point operations
Multiplication based on addition operations
Division takes 1-2 instructions per bit of accuracy
Multiplication can consume much dynamic power
Truncate constants for power savings
56%
Multiplier used in TI C64 processors
[Han, Evans & Swartzlander, 2005]

6. Generating Sinusoidal Waveforms
Sample continuous-time cosine signal at rate fs
Discrete-time fixed frequency 0 = 2  f0 / fs
Example: f0 = 1200 Hz and fs = 8000 Hz, 0 = 3/10 
Discrete-time realization drops fs term in front of cosine
Math library call to cos function in C
Uses double-precision floating-point arithmetic
No standard in C for internal implementation
Generally meant for high-accuracy desktop calculations
Call to gsl_sf_cos_e in GNU scientific library 1.8
20 multiply, 30 add, 2 divide, 2 power calculations/output

7. Generating Sinusoidal Waveforms
Difference equation with input x[n] and output y[n]
y[n] = (2 cos 0) y[n-1] - y[n-2] + x[n] - (cos 0) x[n-1]
From inverse z-transform of z-transform of cos(0 n) u[n]
Impulse response gives cos(0 n) u[n]
2 multiplications and 3 adds per output value
Buildup in error as n increases due to feedback
Lookup table – pre-compute samples offline
Discrete-time frequency 0 = 2  f0 / fs = 2  N / L
All common factors between integers N and L removed
 = 2  k = 2  (N / L) n → n = L → store L samples
Entries in either floating-point or fixed-point format
Table would contain N periods of the cosine
Initial conditions are all zero

8. Generating Sinusoidal Waveforms
Signal quality vs. implementation complexity in generating cos(0 n) u[n] with 0 = 2  N / L
Method
MACs/ sample
ROM (words)
RAM (words)
Quality in floating pt.
Quality in fixed point
C math library call 30 22 1
Second Best
N/A
Difference equation 2 3
Worst
Lookup table L
Best
MAC Multiplication-accumulation RAM Random Access Memory (writeable) ROM Read-Only Memory

9. Discrete-Time Filters
Finite impulse response (FIR) filter
Impulse response h[k] has finite extent k = 0,…, M-1
x[k-1]
x[k]
z-1
z-1 …
z-1 …
h[0]
h[1]
h[2]
h[M-1] S
y[k]
Discrete-time convolution

10. Discrete-Time Filters
Infinite impulse response (IIR) filter
Biquad building block: 2 poles and 0-2 zeros
Generally, coefficients a1, a2, b0, b1, b2 are real-valued 
x[k]
Unit Delay
v[k-1]
v[k-2]
v[k] b2 a1 a2 b1
y[k] b0
Biquad is short for biquadratic− transfer function is ratio of two quadratic polynomials

11. Discrete-Time Filters
FIR Filters
IIR Filters
Implementation complexity (1)
Higher
Lower (sometimes by
factor of four)
Minimum order design
Parks-McClellan (Remez exchange) algorithm (2)
Elliptic design algorithm
Stable?
Always
May become unstable when implemented (3)
Linear phase
If impulse response is symmetric or anti-symmetric about midpoint
No, but phase may made approximately linear over passband (or other band)
(1) For same piecewise constant magnitude specification (2) Algorithm to estimate minimum order for Parks-McClellan algorithm by Kaiser may be off by 10%. Search for minimum order is often needed. (3) Algorithms can tune design to implementation target to minimize risk

12. Discrete-Time Filters
Keep roots computed by filter design algorithms
Polynomial deflation (rooting) reliable in floating-point
Polynomial inflation (expansion) may degrade roots
Choice of IIR filter structure matters
Direct form IIR structures expand zeros and poles, and may become unstable for large order filters (order > 12)
Cascade of biquads expands zeros and poles in each biquad
Minimum order design not always most efficient
Efficiency depends on target implementation
Consider power-of-two coefficient design
Efficient designs may require search of ∞ design space

13. Halftime: AUB Summer 2005 EECE 503 Real-Time DSP Lab
Embedded digital systems
Generating sinusoidal waveforms
Discrete-time filters
Multicarrier equalizers
Image halftoning algorithms
Conclusion

14. Channel Equalization Channel degrades transmitted signal
Nonlinear distortion, e.g. amplitude nonlinearities
Linear distortion, e.g. convolution by channel impulse response
Additive noise, e.g. thermal (Gaussian) and impulsive
Equalization compensates linear distortion
Spreading/attenuation in time
Magnitude/phase distortion in frequency
Transmitter
Channel
Receiver
Equalizer
Message bit stream
Received bit stream

15. Multicarrier Modulation
Divide channel into narrowband subchannels
Discrete multitone modulation
Baseband transmission based on fast Fourier transform (FFT)
Each subchannel carries single-carrier transmission
Standardized for digital subscriber line (DSL) communication
channel
carrier
magnitude
subchannel
Subchannels are 4.3 kHz wide in DSL systems
frequency

16. Equalization in DSL receivers increases bit rate by 10x
Channel Equalization nk
Equalizer
Shortens channel impulse response (time domain eq.)
Compensates phase/ magnitude distortion (freq. domain eq.)
Single carrier system – g is scalar constant
FIR filter w performs time and frequency domain equalization
Multicarrier system – g is FIR filter of length n+1
Time domain equalizer (w) then FFT & freq. domain equalizer
Channel
Equalizer xk yk rk ek h w + + +
Training signal -
Ideal Channel
Receiver generates xk
z- g
Discretized Baseband System
Equalization in DSL receivers increases bit rate by 10x

17. Multicarrier Equalization
Maximum shortening SNR time domain equalizer
Minimize energy leakage outside shortened channel length
For each position of window  [Melsa, Younce & Rohrs, 1996]
Cholesky decomposition of B leads to optimal eigensolution
Computationally-intensive: O(Lw3)
Floating-point multiplications/divisions
Restricts TEQ length to be less than n+1
channel impulse response
effective channel impulse response
n+1 samples

18. Time Domain Equalizer Design
Bit Rate (Mbps)
TEQ length of 17
Data rates averaged over eight standard DSL test lines
[Martin et al., 2006]
Training complexity in log10(multiply-add operations)
Most efficient floating-point versions of algorithms used

19. Time Domain Equalizer Design
Unified framework [Martin et al., 2006]
A and B are square (Lw  Lw) and depend on choice of 
Constraint prevents trivial non-practical solution w = 0
Find eigenvector for largest generalized eigenvalue
Formulation
Power method
Alternating
Lagrangian
Iterative Methods
division-free
20 iterations to converge for 17-tap MSSNR TEQ design

20. Digital Image Halftoning
For display on devices with fewer bits of gray/color resolution than original image
Grayscale: 8-bit image to 1-bit image
Color: 24-bit RGB image to 12-bit RGB display
Produces artifacts
Original Image
b(m)
x(m)
Threshold at Mid-Gray
Each pixel in original image is 8-bit unsigned intensity in [0, 255]
For display, 0 is black and 255 is white

21. Quantization with Feedback
Consider 4-bit data on 2-bit display (unsigned)
Feedback quantization error
For constant input 1001 = 9
Average output value
¼ ( ) = 1001
4-bit resolution at DC !
Noise shaping
Truncating from 4 to 2 bits increases noise by ~12dB
Feedback removes noise at DC & increases HF noise
1 sample
delay
Input signal
words
To display device 4 2
Adder Inputs Output Time Upper Lower Sum to display
Added noise f
12 dB
(2 bits)
Periodic

22. Error Diffusion Halftoning
Quantize each pixel
Diffuse filtered quantization error to “future” pixels
Original
Halftone
difference
threshold + _
e(m)
b(m)
x(m)
compute error
shape error
u(m)
current pixel
Halftone Spectrum
3/16
7/16
5/16
1/16
[Floyd & Steinberg, 1976]
error filter weights

23. Error Diffusion Halftoning
Artifact
Model
Compensation
Added Complexity
Sharpening
Linear
Sharpness control
1 multiplication and 1 addition per pixel
False textures
Nonlinear
Deterministic bit flipping quantization
1 comparison per pixel
Deterministic bit flipping quantizer (DBF) [Damera-Venkata & Evans, 2001]
Thresholds input to black (0) or white (255)
Flip quantized value about mid-gray (128)
Reduces false textures in mid-grays
Implemented with two comparisons
DBF(x)
255 x1
128 x2 x

24. Plots for ideal lowpass H()
Sharpness Control
Model quantizer as gain plus noise [Kite, Evans & Bovik, 1997]
Signal transfer function models sharpening Ks ≈ 2 for Floyd-Steinberg
Noise transfer function models noise-shaping Kn = 1 w w1
-w1 2
Pass low and enhance high frequencies w w1
-w1 1
Pass high frequency noise
Ks = 2
Plots for ideal lowpass H()

25. Sharpness Control
Adjust by threshold modulation [Eschbach & Knox, 1991]
Scale image by gain L and add it to quantizer input
Flatten signal transfer function [Kite, Evans & Bovik, 2000] + _
e(m)
b(m)
x(m)
u(m) L

26. Results
Floyd-Steinberg
Original
DBF quantizer
Unsharpened

27. Conclusion Processor architecture
Decrease data sizes to reduce on-chip memory usage and increase data bus efficiency
Truncate multiplicand constants to reduce power
Compute signal values by recursion or lookup table
Algorithm design
Keep offline design results in full precision until end
Order of calculations matters in implementation
Exploit problem structure in developing fixed-point algorithms
Linearize nonlinear systems to leverage linear system methods
Many other ways to reduce complexity exist

28. Invitations Panel discussion on graduate studies
Tomorrow (Wednesday) 1:30 – 2:30 pm in this room (RCR)
Panelists: Prof. Zaher Dawy (AUB), Prof. Imad El-Hajj (AUB) and Prof. Brian Evans (UT Austin)
IEEE Workshop on Signal Processing Systems
Early October 2011
Short walk from the AUB campus
Organizers include Prof. Magdy Bayoumi (Univ. of Louisiana at Lafayette), Prof. Brian Evans (UT Austin), Dean Ibrahim Hajj (AUB) and Prof. Mohammad Mansour (AUB)

29. Thank You!

30. Digital Signal Processors
DSP Processor Market
Market
~1/3 of $25B embedded digital signal processing market
2007 cholesterol lowering Pzifer Lipitor sales: $13B
Applications (2007)
Annual Revenue
Share
Source: Forward Concepts
Source: Forward Concepts

31. Screening (Masking) Methods
Introduction
Screening (Masking) Methods
Periodic thresholds to binarize image
Periodic application leads to aliasing (gridding effect)
Clustered dot screening is more resistant to ink spread
Dispersed dot screening has higher spatial resolution
Blue larger masks (e.g. 1” by 1”)
Clustered dot mask
Dispersed dot mask
Threshold Lookup Table
index

32. Linear Gain Model for Quantizer
Extend sigma-delta modulation analysis to 2-D
Linear gain model for quantizer in 1-D [Ardalan and Paulos, 1988]
Linear gain model for grayscale image [Kite, Evans, Bovik, 1997]
Error diffusion is modeled as linear, shift-invariant
Signal transfer function (STF): quantizer acts as scalar gain
Noise transfer function (NTF): quantizer acts as additive noise {
us(m) Ks
Ks us(m)
Signal Path
u(m)
b(m)
n(m)
un(m)
Simple linear equalizers are easy to implement but enhance noise
Complex equalizers such as the DFE is required (how many taps has an DFE ?)
The computational complexity of DFE increases very fast with bit rate
un(m) + n(m)
Noise Path

33. Spatial Domain Original Image Threshold at Mid-Gray
Dispersed Dot Screening
Clustered Dot Screening
Floyd Steinberg Error Diffusion
Stucki Error
Diffusion

34. Magnitude Spectra Original Image Threshold at Mid-Gray
Dispersed Dot Screening
Clustered Dot Screening
Floyd Steinberg Error Diffusion
Stucki Error
Diffusion

35. Human Visual System Modeling
Contrast at particular spatial frequency for visibility
Bandpass: non-dim backgrounds [Manos & Sakrison, 1974; 1978]
Lowpass: high-luminance office settings with low-contrast images [Georgeson & G. Sullivan, 1975]
Exponential decay [Näsäsen, 1984]
Modified lowpass version [e.g. J. Sullivan, Ray & Miller, 1990]
Angular dependence: cosine function [Sullivan, Miller & Pios, 1993]

36. Linear Gain Model for Quantizer
Analysis and Modeling
Linear Gain Model for Quantizer
Best linear fit for Ks between quantizer input u(m) and halftone b(m)
Stable for Floyd-Steinberg
Can use average value to estimate Ks from only error filter
Sharpening: proportional to Ks [Kite, Evans & Bovik, 2000]
Value of Ks: Floyd Steinberg < Stucki < Jarvis
Weighted SNR using unsharpened halftone
Floyd-Steinberg > Stucki > Jarvis at all viewing distances
Image
Floyd
Stucki
Jarvis
barbara
2.01
3.62
3.76
boats
1.98
4.28
4.93
lena
2.09
4.49
5.32
mandrill
2.03
3.38
3.45
Average
3.94
4.37