Symbolic analysis and design of communication systems using

computer algebra systems

Prof. Dr Miroslav Lutovac Dr Dejan Tošić

School of Electrical Engineering at the University of Belgrade, Serbia

Overview

• Get back to basic understanding• Numeric vs. Symbolic Computation• Computer as a symbol processor• Schematic as a symbolic object• Programs as knowledge repositories• A step by step example: QAM• Benefits from symbolic techniques

James Kaiser - 50 years of SP

It has become so easy to do so much computation using computers that people will press keys on the keyboard without thinking what they are doing

It's so easy to generate a tremendous amount of garbage that you've got to understand what it is you're doing

Fifty Years of SP (1998), page 54

James Kaiser: “back to basics”

So it is very important that we get back to basic understanding, get a much better grounding of what science underlies the phenomenon we are looking at

I mean, this world is not an ideal world It's time-varying and nonlinear

Fifty Years of SP (1998), page 54

James Kaiser: “understand tools”

Young people or anybody, really who are using these tools have got to thoroughly understand what assumptions underlie the tool that they are using

That will tell them what they can expect to get out

Fifty Years of SP (1998), page 54

Numerical ambiguity: 0.3 - 0.1 == 0.2?

>> a = 0.3-0.1a = 0.2000>> b = 0.2b = 0.2000>> a==bans = 0 0.3 – 0.1 ≠ 0.2

MATLAB Command Window

Why is 0.3-0.1 ≠ 0.2 ?

>> sym(0.8-0.6,'d')ans = .20000000000000006661338147750939>> sym(0.2,'d')ans = .20000000000000001110223024625157>> sym(0.3-0.1,'d')ans = .19999999999999998334665463062265>> sym(0.6-0.4,'d')ans = .19999999999999995559107901499374

MATLAB Command Window

Numeric vs. Symbolic Computation

>> (2/10)==(1-8/10)ans = 0

>> sym(2/10)==sym(1-8/10)ans = 1

>> a=1; a=a-0.2; a=a-0.2; a=a-0.2; a=a-0.2; a=a-0.2a = 5.5511e-017

a ≈ 0, a ≠ 0

Numeric is false

Symbolic is true

Numeric simulation might fail

Symbolic computation finds exact solution

Algebraic loop• Symbolic analysis of systems is

inherently immune to the problem imposed by algebraic loops occurring when two or more blocks with direct feed-through of their inputs form a feedback loop

• Numeric simulations of algebraic loops considerably reduce the speed of a simulation and may be unsolvable

• Symbolic simulation successfully and accurately computes the required response;it finds the exact solution

Computer as a symbol processor

• Computers have become recognized as symbol processors (Oppenheim and Nawab 1992)

• Program can be viewed as a set of instructions for manipulating symbols

• Numbers are only one of the kinds of symbols that the computer can handle

Schematic as a symbolic object System model is a symbolic object It contains all details for drawing, symbolic

solving, simulating, and implementing:1. Analyze the schematic as the symbolic object2. Identify symbolic system parameters3. Knowledge embedded in

the schematic object can be used to generate implementation code or to derive transfer function

From schematic to system property

system = { {"Multiplier", {{6, 0}, {6, 3}}, k1}, {"Delay", {{4, 5}, {4, 7}}, 1}, {"Adder",{{7,8},{8,5},{9,8},{8,9}},{1,1,2,0}}, {"Input", {0, 8}, "X"}, {"Output", {9, 8}, "Y1L"},..., {"Line", {{6, 8}, {7, 8}}} }

k1

k1

k0

b

z1

z1

z1

k2

k2

b

z1

z1

b

z1

z1

z1 z1

k3

k3

b

z1

z1

b

z1

z1

z1 z1

XY1L

Y1H

Y2L

Y2H

Y3L

Y3H

1. Find transfer function

2. Simplify expression

s = H1*(H1/. z->1/z) + H2*(H2/. z->1/z) //FullSimplify

Computer as intelligence amplifier

• Symbol processor with the appropriate programs is usable on a much wider range of tasks, such as intelligence amplifier or augmenting our ability to think

• Programming has become a task of knowledge accumulation telling the computer what to know, when to use, and how to apply the knowledge in solving problems

What to know - How to apply WHAT TO KNOW: symbolic object that contains

a procedure for automated generation of the schematic for an arbitrary number of parts

HOW TO APPLY: draw system, solve symbolically, simulate, and implement system

1. Automatically generate system parameters2. Automatically generate schematic

with symbolic or numeric parameters3. Solve symbolically: find the transfer function, impulse

response, or property of the system from the schematic4. Automatically generate implementation code5. Simulate for specified symbolic parameter values

What to know - When to use WHAT TO KNOW: symbolic object that contains

a procedure for automated generation of the schematic for an arbitrary number of parts

WHEN TO USE: 1. When Laplace or z-transform cannot be found2. When numeric computations fail3. When symbolic expressions have a

large number of parameters4. When derivation by hand is

very time consuming and difficult5. When symbolic optimization can reduce the

number of parameters used in numeric optimization

Knowledge repositories

• Programs are viewed as knowledge repositories

• Programs should be written to communicate …

• … not simply to compute

0 2 4 6 8 10 12 14 16 18 20 22 24

01234567891011

aK

z1

z1

Programs as knowledge repositories1. draw basic part of system

0 2 4 6 8 10 12 14 16 18 20 22 24

01234567891011

a0

X

z1

0 2 4 6 8 10 12 14 16 18 20 22 24

01234567891011

Xd

m

YeYf

2. draw input

3. draw output

(* generate schematic by replicating the basic part *)numberOfStages = 7;adaptiveSystem = TranslateSchematic[...adaptiveSystem = Join[adaptiveSystem, ...Do[adaptiveSystem = Join[adaptiveSystem,... aK -> ToExpression["a"~StringJoin~ ... {k, numberOfStages}];

4. write code

5. Save as function, add knowledge of a system

{schematicSpec, inps, outs} = SchematicFunction[params, …{x0, y0}, options]

Automated drawing of systemsnumStages = 3

p = UnitSymbolicSequence[numStages + 1, k, 0]

parameterSymbols = Join[{b}, p] // Flatten

{hsSystem, inpCoordsHS, outCoordsHS} = HighSpeedIIR3FIRHalfbandFilterSchematic[parameterSymbols];

ShowSchematic[hsSystem]

k1

k1

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XY1L

Y1H

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Y2H

Y3L

Y3H

Invoke from the knowledge repository

Programs written to communicate

k1

k1

k0

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z1

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k2

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Y3L

Y3H

system = { {"Multiplier",{{6,0},{6,3}},k1}, {"Delay",{{4,5},{4,7}},1}, {"Adder",{{7,8},...,{1,1,2,0}}, {"Input",{0,8},"X"},..., {"Line",{{6,8},{7,8}}}}

Out[76]= k2 1 k1 k3 k1 k31 k11 k3 k3

k1

k1

k0

b

z1

z1

z1

k2

k2

b

z1

z1

b

z1

z1

z1 z1

k3

k3

b

z1

z1

b

z1

z1

z1 z1

XY1L

Y1H

Y2L

Y2H

Y3L

Y3H

Solve: find transfer function of all outputs {Y3L/X, Y3H/X}

1z5b z25 k0b5 b4 k1z5 b4 z2 b3 k1k2z2 b3k2k3z2 4 b3 k1z3 b5k1z3 b2k2z3 b2 k1k2k3z3 10 b3z4 3 b2 k1k2z4

2 b4k1k2z4 bk1k3z4 3b2 k2k3z4 2 b4k2k3z4 6 b2k1z5 4 b4 k1z5 2 bk2z5 3b3k2z5 k3z5 2 bk1k2k3z53 b3k1k2k3z5 10 b2 z6 3 bk1k2z6 6 b3k1k2z6 b5k1k2z6 k1k3z6 4 b2 k1k3z6 3 bk2k3z6 6b3 k2k3z6b5k2k3z6 4 bk1z7 6 b3 k1z7 k2z7 6 b2 k2z7 3 b4k2z7 5bk3z7 k1k2k3z7 6 b2k1k2k3z7 3 b4 k1k2k3z75 bz8 k1k2z8 6 b2k1k2z8 3b4 k1k2z8 4 bk1k3z8 6 b3 k1k3z8 k2k3z8 6 b2k2k3z8 3 b4k2k3z8 k1z94 b2k1z9 3 bk2z9 6b3 k2z9 b5k2z9 10 b2k3z9 3 bk1k2k3z9 6b3 k1k2k3z9 b5k1k2k3z9 z10 2 bk1k2z103 b3k1k2z10 6 b2 k1k3z10 4 b4k1k3z10 2 bk2k3z10 3 b3k2k3z10 bk1z11 3 b2 k2z11 2b4k2z11 10 b3k3z113 b2k1k2k3z11 2b4 k1k2k3z11 b2k1k2z12 4 b3 k1k3z12 b5 k1k3z12 b2k2k3z12 b3 k2z13 5 b4k3z13 b3k1k2k3z13 b4 k1k3z14 b5k3z15

Proving the property of the system

1)/1()()/1()( 2211 zHzHzHzH

s = H1*(H1/. z->1/z) + H2*(H2/. z->1/z) //FullSimplify

1z5b z25 k0b5 b4 k1z5 b4 z2 b3 k1k2z2 b3k2k3z2 4 b3 k1z3 b5k1z3 b2k2z3 b2 k1k2k3z3 10 b3z4 3 b2 k1k2z4

2 b4k1k2z4 bk1k3z4 3b2 k2k3z4 2 b4k2k3z4 6 b2k1z5 4 b4 k1z5 2 bk2z5 3b3k2z5 k3z5 2 bk1k2k3z53 b3k1k2k3z5 10 b2 z6 3 bk1k2z6 6 b3k1k2z6 b5k1k2z6 k1k3z6 4 b2 k1k3z6 3 bk2k3z6 6b3 k2k3z6b5k2k3z6 4 bk1z7 6 b3 k1z7 k2z7 6 b2 k2z7 3 b4k2z7 5bk3z7 k1k2k3z7 6 b2k1k2k3z7 3 b4 k1k2k3z75 bz8 k1k2z8 6 b2k1k2z8 3b4 k1k2z8 4 bk1k3z8 6 b3 k1k3z8 k2k3z8 6 b2k2k3z8 3 b4k2k3z8 k1z94 b2k1z9 3 bk2z9 6 b3 k2z9 b5k2z9 10 b2k3z9 3 bk1k2k3z9 6b3 k1k2k3z9 b5k1k2k3z9 z10 2 bk1k2z103 b3k1k2z10 6 b2 k1k3z10 4 b4k1k3z10 2 bk2k3z10 3 b3k2k3z10 bk1z11 3 b2 k2z11 2b4k2z11 10 b3k3z113 b2k1k2k3z11 2b4 k1k2k3z11 b2k1k2z12 4 b3 k1k3z12 b5 k1k3z12 b2k2k3z12 b3 k2z13 5 b4k3z13 b3k1k2k3z13 b4 k1k3z14 b5k3z15

Deriving new propertys = H1*(H1/. z->1/z) + H2*(H2/. z->1/z) //FullSimplify

Solve[s == 1, k0]

Out[57]= k0 121 k121 k221 k32

num3 = Numerator[h3L//Together]/. z -> -1Solve[num3==0, k2]

Out[76]= k2 1 k1 k3 k1 k31 k11 k3 k3

Deriving design equations

Out[57]= k0 121 k121 k221 k32

Out[76]= k2 1 k1 k3 k1 k31 k11 k3 k3

K50 121 K5121 K5221 K532

K52 1 K51 K53 K51 K531 K51 K53 K51 K53

Generating implementation code

DiscreteSystemImplementation[hsSystem,"hsf"]

{{Y9p8, Y9p0, Y31p0}, {Y4p5, Y4p3, Y2p8}} =

hsf[{Y0p8},{Y4p7, Y4p5, Y28p0},{b, k0, k1, k2, k3}] is the template for calling the procedure.

The general template is {outputSamples, finalConditions} = procedureName[inputSamples, initialConditions, systemParameters]. See also: DiscreteSystemImplementationProcessing

1. Output variables

2. Input variables

3. System parameters4. Usage

Get information about implementation procedure

??implementationProcedure

implementationProcedure[dataSamples_List,initialConditions_List,systemParameters_List] :=

Module[{Y0p10,Y4p9,Y4p3,a2,a3,b1,b2,b3},{a2,a3,b1,b2,b3}=systemParameters;{Y0p10}=dataSamples;{Y4p9,Y4p3}=initialConditions;

Y3p0=b3 Y0p10;Y3p4=b2 Y0p10;Y3p10=b1 Y0p10;Y4p5=Y3p4+Y4p3;Y8p10=Y3p10+Y4p9;Y5p0=a3 Y8p10;Y5p6=a2 Y8p10;Y4p1=Y3p0-Y5p0;Y4p7=Y4p5-Y5p6;{{Y8p10},{Y4p7,Y4p1}}]

1. Variables

2. Input variables

3. Initial conditions4. Code

2. Transfer function

3. Time response

1. Symbolic parameter

Transferfunction matrix

of MIMO system

Simulation with

symbolic system

parameters

Transfer function

Symbolic simulation

Symbolic processingnumberInSamples = 20;

inputSequence = UnitImpulseSequence[numberInSamples];

eqns = DiscreteSystemImplementationEquations[hsSystem];initialConditions = 0*eqns[[2]]systemParameters = eqns[[3]]

{outputSeq, finalCond}=DiscreteSystemImplementationProcessing[inputSequence, initialConditions, systemParameters, hsf];

Out[85]=

b51 b2k0b7 9 b2k2 k156 b7 k2 k3 b280 b220 b2 b2 9 k2 k3Each element

of the output sequence is a symbolic expression

Response in time domain

p={b→9/16,k0→0.24000685,k1→2.37428,k2→-0.54068,k3→0.1093268}

y=InverseZTransform[hsSystem /. p, z, n]

Use z-transform (if it exists)

How to synthesize a discrete system?

1. For known transfer function H(z) = ( 1 + 2 z -1 + z -2 ) / ( 1 + ½ z -2 )

create schematic of the system {schematic, {inpCoord}, {outCoord}} =TransposedDirectForm2IIRFilterSchematic[{{1,2,1},{0,1/2}}];

2. Add input element and output elementsystem = Join[schematic,{{"Input",inpCoord,X}, {"Output",outCoord,Y}}]

3. Draw the block-diagramShowSchematic[system]

Invoke from the knowledge repository

Discrete systems analysis:Find response from the schematic

4. Compute transfer function from the schematic{tfMatrix, systemInp, systemOut} = DiscreteSystemTransferFunction[system];tf = tfMatrix[[1, 1]];

5. Input signal represented by a formulasineSignal = Sin[n/5];

6. Find output signalsineTransform = ZTransform[sineSignal, n, z];response = InverseZTransform[sineTransform*tf,z,n]

8 Cos11022n2 Cosn

2Sin1

5 2 Sin1n

5 Sin1n

5 2 1

2 n2Sin1

5 4 Sin2

5Sinn

2

5 4 Cos25

Discrete systems analysisusing symbolic processing

1. Generate a code that implements the systemDiscreteSystemImplementation[system, "imp"];

2. Compute input sequence whose elements can be symbols, numbers, or formulasinSeq = UnitSineSequence[8, 1/(10 π), 0];

3. Process the input sequence with the code{outSeq,finals} = DiscreteSystemImplementationProcessing[inSeq,{0,0},{},imp];

142 Sin1

5 Sin2

5 4 Sin3

5 2 Sin4

5 8 Sin1 Sin6

5

The seventh element of the output sequence

is not a number; it is an expression

Comparing multirate realizations

outClassic

outSeq

In[88]:=SameQoutClassic, outSeq

Out[88]=True

Quadrature Amplitude Modulation

A step by step example

What is QAM?

• Quadrature Amplitude Modulation (QAM) is a widely used method for transmitting digital data over bandpass channels

• The simulation of a simplified and idealized QAM system follows

Read-in the knowledge

Generate the transmitter part

Generate the demodulator part

Generate the filter part

Generate the complete system

Generate the implementation code

Generate the input sequences

20 40 60 80 100

-1

-0.5

0.5

1

Process the input sequences with the

system

Simulate the input sequences with the

system

Miscellaneous examples

Hilbert Transformer

Hilbert Transformer in QAM

Input and Output Sequences

0.002 0.004 0.006 0.008 0.01 0.012Times

-1

-0.5

0.5

1

0.002 0.004 0.006 0.008 0.01 0.012Times

-1

-0.5

0.5

1

Better output with Hilbert transformer

20 40 60 80 100

-0.6

-0.4

-0.2

0.2

0.4

0.6

Classic filter produces this

-4000 -2000 0 2000 4000Frequency Hz0

0.1

0.2

0.3

0.4

0.5

murtcepS

Output of modulatorSystem

-4000 -2000 0 2000 4000Frequency Hz0

0.2

0.4

0.6

0.8

1

murtcepS

Input to modulatorSystem

-4000 -2000 0 2000 4000Frequency Hz0

0.2

0.4

0.6

0.8

murtcepS

Output of modulatorSystem

Spectra of QAM Signals

Complex signal

Modulated signal

Input signal

Amplitude Modulation

Nonlinear systems

Can we find the output signal as a closed-form expression in terms of the sample index?

Nonlinear systems: Symbolic response

1. Draw the schematic2. Automatically generate a code that implements the system

DiscreteSystemImplementation[system, "implement"];3. Compute the successive output values

{{y2}, {d2}} = implement[{1,10}, {d1}, {}];{{y3}, {d3}} = implement[{1,10}, {d2}, {}];

4. Eliminate the initial states and find the relation between the output sampleseqns = Reduce[{y[n-1]==y2, y[n]==y3}, {d1}];

5. Find the recurrence equationreducedEqn =(15 y[-1 + n] == -10 + 16 y[n]);

6. Load knowledge for solving recurrence equations<<DiscreteMath`RSolve`

7. Find a closed-form solutionRSolve[{reducedEqn,y[0]==0},y[n],n];yn 101 15

16n

Nonlinear systems: Optimization

• Example: find the number of samples after which the output sequence reaches some value, say b

• Solve[y[n] == 10*b, n];• Solution:

• Verification:

n Log1 bLog16

15

Adaptive System

Finding Adapted Coefficients

Symbolic Response of the Unknown System

• desiredSignalSymbolic = DiscreteSystemSimulation[unknownSystem,inputSignal]

• {{-0.0026 b0}, {-0.1111 b0 - 0.0026 b1}, {0.0751 b0 - 0.1111 b1 - 0.0026 b2}, {0.05 b0 + 0.0751 b1 - 0.1111 b2 - 0.0026 b3}, {-0.0517 b0 + 0.05 b1 + 0.0751 b2 - 0.1111 b3 - 0.0026 b4}, … }={{0.000013}, {0.0006075}, {0.0015865}, {-0.013902}, ...}

Automatic Gain Control

Nonlinear

Power

System output

Scaled Signal

Gain

Power

Algorithm development

Efficient method for approximating the reciprocal using a modified Newton-Raphson iteration

yn 1 1 b x2nx

Algorithm development1. Draw the schematic of algorithm2. Automatically generate a code that implements the system

DiscreteSystemImplementation[systemNR, "implementNR"];3. Compute the successive output values

{{y2}, {d2}} = implementNR[{x,2}, {d1}, {}];{{y3}, {d3}} = implementNR[{x,2}, {d2}, {}];

4. Eliminate the initial states and find the relation between the output sampleseqns = Reduce[{y[n - 1] == y2, y[n] == y3}, {d1}];

5. Find the recurrence equationreducedEqn = (x*y[n-1]^2 == 2*y[n-1] - y[n]);

6. Load knowledge for solving recurrence equations<<DiscreteMath`RSolve`

7. Find a closed-form solutionsol = RSolve[{reducedEqn,y[0]==b},y[n],n];

yn 1 1 b x2nx

Algorithm development: Optimization of initial guess

• Example: algorithm for implementation of an efficient method for approximating the reciprocal using a modified Newton-Raphson iteration

• Find the initial guess to minimize the error of the approximate reciprocal in terms of the given number x, the initial guess b and the number of iterations n:FindRoot[e[n] + 1/2^16 == 0, {b,2}];

• Solution: b = 1.98923

• The error is smaller than 2-16 for x over the range 0.01 < x < 1

Conclusion

• Contemporary trends to use very sophisticated algorithms combine expertise in many areas, such as communications engineering, computer science, ICT, and signal processing

• Current symbolic computation environments are powerful in doing symbolic and mixed symbolic-numeric mathematics for technical computing

Conclusion (2)

• Programs provide knowledge about design and employ the knowledge in symbolic manipulation:

a) automated generation of schematic objects and the corresponding implementation codes

b) derivation of the transfer function, system properties and time response

c) symbolic optimization

Conclusion (3)

• Superiority of symbolic computation against numerical computation was shown by

a) the example system with an algebraic loop; CAS yielded the exact solution while the traditional numeric approach failed

b) the closed-form solution of a nonlinear LMS subsystem

c) deriving the analytic expression for the error of the Newton-Raphson iteration

Conclusion (4)

• Benefits of symbolic methods were highlighted from the viewpoint of

a) Academia (derivation of time and frequency response, proving system properties)

b) Industry (QAM, Hilbert transformer, LMS algorithm, verification of realizations, design alternatives in multirate systems)

Examples and documentationhttp://library.wolfram.com/infocenter/TechNotes/4814/

http://www.schematicsolver.com

Further reading

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