digital signal processing -...

VLSI Digital Signal Processing Systems

Introduction to Digital Signal Processing Systems

Lan-Da Van (范倫達), Ph. D.

Department of Computer Science

National Chiao Tung University Taiwan, R.O.C.

Spring, 2020

[email protected]

http://www.cs.nctu.edu.tw/~ldvan/


Lan-Da Van VLSI-DSP-1-2

Outlines

Introduction

DSP Algorithms

DSP Applications and CMOS IC’s

Representations of DSP Algorithms

Conclusion



Why Use Digital Signal Processing?

Robust to temperature and process variations

Controlled better to accuracy

Noise/interference tolerances

Mathematical representation

Programming capability



Common System Configuration

Multimedia-Communication-Biomedical

Applications

VLSI Signal Processing Library Processor Software



VLSI Signal Processing System Design Spectrum (1/2)

Computer arithmetic

Adder

Multiplier

Inverse square root

Division

Digital filter

FIR/IIR

Multidimensional filter

Adaptive digital filter

LMS/DLMS (Delay LMS) based

RLS based

Transform

Multiplier-accumulator based

Recursive-filter based

ROM-based: DA, CORDIC

Butterfly based

Processor

General purposed processor

DSP processor

Reconfigurable computing

processor

Machine Learning

Independent Component

Analysis (ICA)

Conventional Neural Network

(CNN)

Recurrent Neural Network

(RNN)

Reinforcement Learning

3D Graphics

Rasterization

Ray Tracing

Photon Mapping



VLSI Signal Processing System Design Spectrum (2/2)

MIMO Detection

Grouped Detection

VBLAST

K-Best

Image Processing

Pattern Recognition

Median Filter

Image Reconstruction

Image Enhancement

Video Processing

• Compression

• Block Matching

• Deblocking filter

Non-numerical operation

Error control coding

Viterbi Decoder

Turbo Code

Polynomial computation

Dynamic programmable



VLSI Signal Processing System Publication Area (But not limited…)

IEEE Access

IEEE Systems Journal

IEEE Trans. on Biomedical Engineering

IEEE Trans. on Circuits and Systems I: Regular Papers

IEEE Trans. on Circuits and Systems II: Express Briefs

IEEE Trans. on Circuits and Systems for Video Technology

IEEE Trans. on Communications

IEEE Trans. on Computer-Aided Design of Integrated Circuits

IEEE Trans. on Computers

IEEE Trans. on Image Processing

IEEE Trans. on Multimedia

IEEE Trans. on Neural Networks

IEEE Trans. on Signal Processing

IEEE Journal of Solid-State Circuits

IEEE Trans. on VLSI Systems

IEEE Trans. on Visualization and Computer Graphics

Proceedings of the IEEE

ACM Trans. on Graphics

Journal of Signal Processing Systems

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

Elsevier Integration - The VLSI Journal



VLSI Signal Processing System Design Space

System Level

Algorithm Level

Architecture Level

Circuit Level

Logic Level

Process Level

Power

Area

PerformanceCost

Test



Outlines

Introduction

DSP Algorithms



Conclusion



DSP Algorithms

Convolution

Correlation

Digital filters

Adaptive filters

Discrete Fourier transform

Source Coding Algorithms Discrete cosine transform

Motion estimation

Huffman coding

Vector quantization

Algorithm: A set of rules for solving a problem in a finite number of steps.



Signals

Analog signal

t->y: y=f(t), y:C, t:C

Discrete-time signal

n->y: y=f(nT), y:C, n:Z

Digital signal

n->y: y=D{f(nT)}, y:Z,n:Z

)3(

)1(

)2(

)1(

2)1110(

2)1000(2)1011(

2)1000(

t n n

y y y

Analog Signal Discrete-Time Signal Digital Signal



LTI Systems

Linear systems

Assume x1(n)->y1(n) and x2(n)->y2(n), where “->” denotes

“lead to”. If ax1(n)+bx2(n)->ay1(n)+by2(n), then the systems is

referred to as “Linear System.”

Homogenous and additive properties

Time-invariant (TI) systems

x(n-n0)->y(n-n0)

LTI systems

y(n)=h(n)*x(n)

Causal systems

y(n0) depends only on x(n), where n<=n0

Stable systems

BIBO



Sampling of Analog Signals

Nyquist sampling theorem

The analog signal must be band-limited

Sample rate must be larger than twice the bandwidth



System-Equation Representation

Impulse/unit sample response

Transfer function / frequency response

Difference equations

State equations

11

0

1)(

)()(

za

b

zX

zYzH

)()1()( 01 nxbnyany

][)( 10 nuabnh n



Convolution & Correlation

Convolution

k

knhkxnhnxny )()()()()(

)()()()( nxnaknxka

k

k

knxkany )()()(

Correlation

)()( knxkhk



Linear Phase FIR Digital Filters

Digital filters are an important

class of LTI systems.

Linear phase FIR filter

)()( nMhnh

0)6()0( bhh

1)5()1( bhh

2)4()2( bhh

3)3( bh

)4()5()6(

)3()2()1()()(

210

3210

nxbnxbnxb

nxbnxbnxbnxbny



IIR Filter Structures

2

2

1

1

2

2

1

10

1)(

zaza

zbzbbzH

1z

1z



Introduction to an Adaptive Algorithm

Widely used in communications, DSP,

biomedical, and control system

Deterministic gradient algorithm

RLS algorithm

Stochastic gradient algorithm

LMS algorithm, DLMS algorithm

Block LMS algorithm

Gradient lattice algorithm



Adaptive Applications

Channel equalizer

System identification

Echo canceller

Noise cancellation

Predictor

Beamformer

Line enhancement

Image enhancement

Machine Learninghttps://en.wikipedia.org/wiki/Least_mean_squares_filter



Notation

)Error: e(n

Factor: μAdaptation

tor: w(n)Weight Vec

tput: d(n)Desired Ou

al: x(n)Input Sign

.5

.4

.3

.2

.1

: .10

.9

.8

.7

.6

MatrixDiagonal

:λEigenvalue

ix: Ration MatrAutocorrel

: NTap Number

ent: MMisadjustm adj



LMS Algorithm

)()()( nnny Txw

TNnxnxnxnwhere )]1( ... )1( )([)( x

TN nwnwnwn )]( ... )( )([)( 110 w

)()()(

)()()(

)()()(

nnnd

nnnd

nyndne

T

T

wx

xw

The error at the n-th time is



LMS Algorithm

))(ˆ(2

1)()1( nJnwnw

)()(2)(ˆ nxnenJ

μe(n)x(n)w(n))w(n 1

w0’, w1, w0

An efficient implementation in software of steepest

descent using measured or estimated gradients, where

the cost function with gradient is square errors.

The gradient of the square of a single error sample

Cost



)()()( nyndne

Summary of LMS Adaptive Algorithm (1960)

)()()( nnny Txw

)()( )()1( nnenn xww



Block Diagram of an Adaptive FIR Filter Driven by the LMS Algorithm



Unitary/Orthogonal Transform (1/4)

Definition: (from Linear Algebra)

Let A be nn matrix that satisfies

IAAAA .

We call A as an unitary matrix if A

has complex entries, and we call A as

an orthogonal matrix if A has real

number.



Why Orthogonal Transformation? (2/4)

Energy conservation

Energy compaction

Most unitary transforms tend to pack a large fraction of the

average energy of signals into a relatively few components

of the transform coefficients.

Decorrelation

When signals are highly correlated, the transform

coefficients tend to be uncorrected (or less correlated).

Information preservation

The information carried by signals are preserved under a

unitary transform.




0 100 200 300 400 500 60050

100

150

200

250The original signal

0 100 200 300 400 500 600-2000

0

2000

4000The DCT coefficients

Source: Lecture of Prof. Dennis Deng




0 5 10 15 20 25

0.98

1

The auto correlation of original signal

0 5 10 15 20 25-0.5

0

0.5

1The auto correlation of DCT coefficients

Source: Lecture of Prof. Dennis Deng



Discrete Fourier Transform (1/9)

DFT

IDFT

1,...,1,0,)()(

1

0

NkWnxkX

N

n

nkN

1,...,1,0,)(1

)(

1

0

NnWkXN

nx

N

k

nkN

nkN

jnkN

Nj

N eWeW

22

,



Fast Fourier Transform (2/9)

The radix-2 algorithm is the most widely used fast

algorithm to compute the DFT.

Without loss of generality, we use an 8-point DFT (N=8)

to illustrate the development of the fast algorithm.

))7()5()3()1((

)6()4()2()0(

)7()5()3()1(

)6()4()2()0(

)()(

642

642

753

642

7

0

kN

kN

kN

kN

kN

kN

kN

kN

kN

kN

kN

kN

kN

kN

knN

n

WxWxWxxW

WxWxWxx

WxWxWxWx

WxWxWxx

WnxkX




Since

8-point DFT => Nearly two 4-point DFT

where represent the DFT of two sequences

kN

kN

jkN

jk

N WeeW 2/)2/(

22

2

2

1,...2,1,0 where),()(

))7()5()3()1((

)6()4()2()0()(

21

32/

22/2/

32/

22/2/

NkkFWkF

WxWxWxxW

WxWxWxxkX

kN

kN

kN

kN

kN

kN

kN

kN

)()( 21 kFandkF

)12()()2()( 21 nxnfandnxnf




One step further: (=>Two 4-point DFT)

An N-point DFT requires N2 complex multiplications. The

number of complex multiplications required by the above

algorithm is as follows.

An 8-point DFT requires 64 complex multiplications.

12/,...,1,0 ),()()( 21 NkkFWkFkX kN

12/,...,1,0 ),()()2/( 21 NkkFWkFNkX kN

kN

jk

NjNk

Nj

NkN WeeeW

2)2/(

2

2/

)8( 40)2/(2 2 NNN

kN

NkN WW 2/

2/2/ and




The 4-point DFT can be decomposed into two 2-point

DFT in a similar way.

where V11(k) and V12(k) represent the DFT of two

sequences.

As before,

)()(

))6()2(()4()0(

)6()2()4()0()(

122/11

4/2/4/

32/2/

22/1

kVWkV

WxxWWxx

WxWxWxxkF

kN

kN

kN

kN

kN

kN

kN

)12()( )2()( 112111 nfnvandnfnv

14/,...,1,0),()()( 122/111 NkkVWkVkF kN

14/,...,1,0),()()2/( 122/111 NkkVWkVNkF kN




A 2-point FFT, such as involves only

real addition

)()( 1211 kVandkV

1,1),4()0()( 12

0211 2

WWxWxkV k

V x W x

V x W x

N

N

110

110

0 0 4

1 0 4

( ) ( ) ( )

( ) ( ) ( )

a

b

A

B-1

WN

Each butterfly requires one complex multiplication and two complex addition




After decimation, the sequence is in a bit-reversed

order

original order decimation1 decimation 2

n2n1n0 n0n2n1 n0n1n2

0 000 0 000 0 000

1 001 2 010 4 100

2 010 4 100 2 010

3 011 6 110 6 110

4 100 1 001 1 001

5 101 3 011 5 101

6 110 5 101 3 011

7 111 7 111 7 111




This FFT algorithm is generally true for any data

sequence of

There are N/2 butterflies per stage and

stages

The number of operations required for an FFT:

(Before simplifying)

Complex multiplication:

Complex addition:

vN 2

NN 2log

NN 2log

N2log



Image/Video Compression

Where coding? Source coding

Channel coding

Source coding benefits Lower bit rate

Less transmission time

Fewer storage data

What kind of loss? Lossless data compression

Lossy data compression

Why can we do compression? Coding redundancy

Inter-sample redundancy (Spatial redundancy)

Inter-frame redundancy (Temporal redundancy)



Source Coding Spectrum

Image Compression

Lossless Loss

Huffman Coding

Shannon Coding

ArithmeticCoding

Predictive Coding

Transform Coding

VQ Coding

Subband Coding



Image Measurement and Evaluation

)/(log10SNR(dB) 2210 nx

)/255(log10PSNR(dB) 2210 n

ly.respective valules,image tedreconstruc and

image orignal thedenote ˆ and where (i,j)xx(i,j)

2

1 1

2]),(),([

1

N

i

N

j

n jixjixN



Discrete Cosine Transform (DCTII)

10 , ]2

)12(cos[)()()(

1

0

N-kN

knnxkkX

N

n

N

1)0( 1 1for ,

2 )( Nk

Nk



2-D DCT-II and IDCT-II

)2

)12(cos()

2

)12(cos(),()()(

2),(

1

0

1

0N

ln

N

kmlkZlk

Nnmx

N

k

N

l

)2

)12(cos()

2

)12(cos(),()()(

2),(

1

0

1

0N

ln

N

kmnmxlk

NlkZ

N

m

N

n

DCT-II

IDCT-II

.0for 1 and 210

and 1 to0 from ranges and , where

jα(j)/)α(

N-nk, l, m

TAXAZ

ZAAX T



How to Decide the Coefficients?

Orthogonal Property

IAAAA TT Parseval’s Theorem: Energy Conservation

1

0

21

0

2)(

1)(

N

k

N

n

kXN

nx



2-D DCT/IDCT Processor

(a)

(b)



Block-Matching Algorithm

1

0

1

0

),(),(),(N

i

N

j

njmiyjixnms pnmpfor ,

)},({min ),( nmsu nm pnmpfor ,

unmv ),(

Rule:



Entropy (1/2)

「熵」是熱力學的概念，由德國物理學家克勞修斯（Rudolf Clausius）在1864年提出。「熵」的被用來度量一個混亂、不確定系統，其混亂或不確定的程度。熵是對某個系統之不確定性或混亂程度的度量方法。

【資訊】是個很抽象的概念，我們常常說資訊很多，或者資訊較少，但卻很難說清楚資訊到底有多少。直到1948 年， Prof. Claude Shannon 提出了information theory (then known as communication theory) 的概念後，才解決了對資訊的量化度量問題。

q

i

ii ppP

1

2 )/1(log)H(

https://blog.xuite.net/metafun/life/69851478-%E8%B3%87%E8%A8%8A%E7%9A%84%E5%BA%A6%E9%87%8F-+Information+Entropy




Entropy (1/2)

這個公式和熱力學的熵的計算方式一樣，故也稱為熵(資訊理論)。從公式可知：當機率越平均時，也就是個事件機率越均勻(意謂：越無法預測，也越模糊)，「不確定性」(uncertainty)最高，【訊息熵】最大。

物理意義:

Information Measurement

Uncertainty Measurement

Surprise Measurement

「資訊熵」與「物理熵」是息息相關的。舉例而言，當我們將1莫耳（mol）理想氣體的體積在恆溫下壓縮一半時，「物理熵（ΔS）」減少了，但由於分子分布的空間縮小，其位置的確定性增高，因此「資訊熵（ΔI）」增加了，如果我們認定「資訊熵」的增加與「物理熵」的減少是相同的（ΔI＝－ΔS），則取得一位元的資訊相當於每莫耳理想氣體溫度上升 1K 所需能量（J／mol．K）。 https://scitechvista.nat.gov.tw/c/s24Z.htm


https://scitechvista.nat.gov.tw/c/s24Z.htm




Huffman Coding (1/2)

1x

2x

6x

5x

4x

3x

8x

7x

Input Probability

2

1

1

64

1

16

1

8

1

4

1

64

1

64

1

64

1 0

1

0

1

0

1

0

1

0

1

0

1

0



Huffman Coding (2/2)

Data Huffman Code Natural Code

1x 1 000

2x 01 001

3x 001 010

4x 0001 011

5x 000001 100

6x 000000 101

7x 000011 110

8x 000010 111

bit 2(4x6)64

1x4

16

1x3

8

1x2

4

1x1

2

1

AvLen deHuffman_Co

bit 3AvLen deNatural_Co

bit 264)(4xlog64

116log

16

18log

8

14log

4

12log

2

1

)(

22222

EntropyxH

bits coding

bits uncodingCr

5.12

3



Outlines

Features:

DSP Algorithms



Conclusion



Moore’s Law

Microns

Source: Sematech

Tr. # (Complexity) Tr. # (Productivity)

10

1

0.01

0.1

10G

100M

10M

1M

100K

10K

1K

1G

1980 1985 1990 20001995 20102005

10

100M

10M

1M

100K

10K

1K

100x

xx

xx x

x

x

Gate Length

Device Complexity

Design Productivity

21%/ year

58% / year

Gap

Increases

The number of transistors per chip doubles every 18 months.

* Cordon Moore: One of the founders of Intel



Moore’s Law

https://en.wikipedia.org/wiki/Moore%27s_law#/media/File:Moore's_Law_Transistor_Count_1971-2018.png

https://en.wikipedia.org/wiki/Moore%27s_law#/media/File:Moore's_Law_Transistor_Count_1971-2018.png



Moore’s Law

https://www.mile.cloud/gcp/nvidia-ceo-moores-law-is-over/

Com

putin

g P

ow

er

https://www.mile.cloud/gcp/nvidia-ceo-moores-law-is-over/



Evolution of Applications



Chronological Table of Video Coding Standards (1/2)

H.261

(1990)

MPEG-1

(1993)

H.263

(1995/96) H.263+

(1997/98)

H.263++

(2000)

H.264

(MPEG-4

Part 10 )

(2002)MPEG-4 v1

(1998/99)

MPEG-4 v2

(1999/00)

MPEG-4 v3

(2001)

1990 1992 1994 1996 1998 2000 2002 2003 2013

MPEG-2

(H.262)

(1994/95)ISO/IEC

MPEG

ITU-TVCEG

H.265

(HEVC)

(2013)



Chronological Table of Video Coding Standards (2/2)

https://www.twblogs.net/a/5c4abacebd9eee6e7e069d9c



http://www.point.tv/products/h265_encoder

Comparison of Video Coding Standards (1/2)



Comparison of Video Coding Standards (2/2)





Block Diagram of H.264/AVC Encoder

Entropy

Coding

Scaling & Inv.

Transform

Motion-

Compensation

Control

Data

Quant.

Transf. coeffs

Motion

Data

Intra/Inter

Coder

Control

Decoder

Motion

Estimation

Transform/

Scal./Quant.-

Input

Video

Signal

Split into

Macroblocks

16x16 pixels

Intra-frame

Prediction

De-blocking

Filter

Output

Video

Signal



3D Graphics System => GPU

Geometry Engine

Raster Engine



Shading Algorithms

Gouraud shading

Per-vertex lighting

Low computation

Not good shading quality

Phong shading

Per-pixel lighting

Huge computation

Smooth and more realistic highlight

Existing Approximate Phong Shading Algorithms

Mixed shading

Subdivision based approximate algorithms

Mixed shading

No pass

Pass

Source: ACM/IEEE


Four Area Networks

From big to small:

Wireless Wide Area Network

Wireless Metro Area Network

Wireless Local Area Network

Wireless Personal Area Network

資料來源：無線都會網路新貴WiMAX標準介紹，http://tech.digitimes.com.tw/ShowNews.aspx?zCatId=134&zNotesDocId=E88A9E150386245D48256F5B00128CAC

https://networksmania.wordpress.com/gallery/images/networking-standards/


http://tech.digitimes.com.tw/ShowNews.aspx?zCatId=134&zNotesDocId=E88A9E150386245D48256F5B00128CAC


Source: UMTS Forum

WAN

LAN

PAN

Low data rate High data rate

Lo

w M

ob

ility

Hig

h M

obili

ty

GSM/GPRS

802.11a/g

ZigBee802.15.4

Bluetooth802.15.1

WiMedia802.15.3a

WiMAX802.16d

MAN

10 Mbps0.1 Mbps 100 Mbps

WCDMA

RFID

802.11n

1 Mbps 1000 Mbps

802.11b

HSPA 3GPP LTE

WiMAX802.16e

WiMAX802.16m


Communication Standards Evolution (1/4)




https://gogonews.cc/article/6231376.html

https://gogonews.cc/article/6231376.html


http://t17.techbang.com/topics/20299-d-link-11-ac-cloud-of-the-worlds-fastest-route-to-experience-carry-out-d-will-overlord-netcom-equipment-d-link-corp?mode=print&page=1



http://www.digitimes.com.tw/iot/article.asp?cat=130&id=0000377870_a7o9e4no6qed6d1ajd2kp




Error Control

Decoder

Digital Communications System

Enabling the transmitted signal to withstand the effects of

various channel impairments, such as noise, interference,

and fading.

Information

SourceSource

EncoderEncrypter

Error Control

EncoderModulator

Channel

Demodula

torInformation

SinkDecrypter

Source

Decoder



Outlines

Features:

DSP Algorithms



Dependence Graph

Data-Flow Graph

Signal-Flow Graph

Block Diagrams

Conclusion



DG of a 3-Tap FIR Filter

Def: A dependence graph is

a direct graph that shows

the dependence of the

computations in an

algorithm. The node in a

DG represent computations

and the edges represent

precedence constraints

among nodes. DG contains

computations for all

iterations in an algorithm

and does not contain delay

elements.



DFG of a 3-Tap FIR Filter

Def: A data flow graph

(DFG) is a collection of

nodes and directed edges.

The nodes represent

computations (or functions

or subtasks) and the

directed edges represent

data path and each edge

has a nonnegative number

of delays associated with it.



SFG of a 3-Tap FIR Filter

Def: A signal flow graph

(SGF) is a collection of

nodes and directed edges.

The nodes represent

computations or tasks. In

digital networks, the

edges are usually

restricted to constant gain

multipliers or delay

elements.



Block Diagram of a 3-Tap FIR Filter

Def: A block

diagram consists of

functional blocks

connected with

directed edges.



Conclusion

Briefly introduced the following:

DSP design issue and design view

DSP algorithms

Overview of DSP applications

Representations of DSP algorithms



References

[1] K. K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation. NY: Wiley, 1999.[2] P. Pirsch, Architectures for Digital Signal Processing. NY: Wiley, 1998.[3] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.[4] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.[5] 連國珍,數位影像處理, 1992.

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