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Symbolic analysis and design of communication systems using computer algebra systems

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Symbolic analysis and design of communication systems using computer algebra systems

Prof. Dr Miroslav Lutovac Dr Dejan TošićSchool of Electrical Engineering at the University of Belgrade, Serbia

- 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

- 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 garbagethat you've got to understand what it is you're doing

Fifty Years of SP (1998), page 54

- 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

- Young people or anybody, really who are using these tools have got to thoroughly understandwhat 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

>> a = 0.3-0.1

a = 0.2000

>> b = 0.2

b = 0.2000

>> a==b

ans = 0

MATLAB Command Window

0.3 – 0.1 ≠ 0.2

MATLAB Command Window

>> 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

>> (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-0.2

a = 5.5511e-017

Numeric is false

Symbolic is true

a ≈ 0, a ≠ 0

- 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

- 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

- System model is a symbolic object
- It contains all details for drawing, symbolic solving, simulating, and implementing:
- Analyze the schematic as the symbolic object
- Identify symbolic system parameters
- Knowledge embedded in the schematic object can be used to generate implementation code or to derive transfer function

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}}} }

1. Find transfer function

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

2. Simplify expression

- 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: 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
- Automatically generate system parameters
- Automatically generate schematic with symbolic or numeric parameters
- Solve symbolically: find the transfer function, impulse response, or property of the system from the schematic
- Automatically generate implementation code
- Simulate for specified symbolic parameter values

- WHAT TO KNOW: symbolic object that contains a procedure for automated generation of the schematic for an arbitrary number of parts
- WHEN TO USE:
- When Laplace or z-transform cannot be found
- When numeric computations fail
- When symbolic expressions have a large number of parameters
- When derivation by hand is very time consuming and difficult
- When symbolic optimization can reduce the number of parameters used in numeric optimization

- Programs are viewed as knowledge repositories
- Programs should be written to communicate …
- … not simply to compute

1. draw basic part of system

5. Save as function,

add knowledge of a system

{schematicSpec, inps, outs} =

SchematicFunction[params, …{x0, y0}, options]

2. draw input

(* generate schematic by replicating the basic part *)

numberOfStages = 7;

adaptiveSystem = TranslateSchematic[...

adaptiveSystem = Join[adaptiveSystem, ...

Do[adaptiveSystem = Join[adaptiveSystem,...

aK -> ToExpression["a"~StringJoin~ ... {k, numberOfStages}];

3. draw output

4. write code

numStages = 3

Invoke from the knowledge repository

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

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

{hsSystem, inpCoordsHS, outCoordsHS} =

HighSpeedIIR3FIRHalfbandFilterSchematic[parameterSymbols];

ShowSchematic[hsSystem]

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}}}}

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

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

Solve[s == 1, k0]

num3 = Numerator[h3L//Together]/. z -> -1

Solve[num3==0, k2]

DiscreteSystemImplementation[hsSystem,"hsf"]

1. Output variables

{{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

2. Input variables

3. System parameters

4. Usage

??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 conditions

4. Code

1. Symbolic parameter

2. Transfer function

3. Time response

Transfer

function matrix

of MIMO system

Transfer function

Simulation with symbolic system parameters

numberInSamples = 20;

inputSequence = UnitImpulseSequence[numberInSamples];

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

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

Each element of the output sequence is a symbolic

expression

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)

- For known transfer functionH(z) = ( 1 + 2 z-1 + z-2 ) / ( 1 + ½z-2 )create schematic of the system{schematic, {inpCoord}, {outCoord}} =TransposedDirectForm2IIRFilterSchematic[{{1,2,1},{0,1/2}}];
- Add input element and output elementsystem = Join[schematic,{{"Input",inpCoord,X}, {"Output",outCoord,Y}}]
- Draw the block-diagramShowSchematic[system]

Invoke from the knowledge repository

- Compute transfer function from the schematic{tfMatrix, systemInp, systemOut} = DiscreteSystemTransferFunction[system];tf = tfMatrix[[1, 1]];
- Input signal represented by a formulasineSignal = Sin[n/5];
- Find output signalsineTransform = ZTransform[sineSignal, n, z];response = InverseZTransform[sineTransform*tf,z,n]

- Generate a code that implements the systemDiscreteSystemImplementation[system, "imp"];
- Compute input sequence whose elements can be symbols, numbers, or formulasinSeq = UnitSineSequence[8, 1/(10 π), 0];
- Process the input sequence with the code{outSeq,finals} = DiscreteSystemImplementationProcessing[inSeq,{0,0},{},imp];

The seventh element of the output sequence is not a number; it is an expression

outClassic

outSeq

Quadrature Amplitude Modulation

A step by step example

- 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

Process the input sequences with the system

Simulate the input sequences with the system

Miscellaneous examples

Better output with Hilbert transformer

Classic filter produces this

Input signal

Modulated signal

Complex signal

Can we find the

output signal as

a closed-form

expression in

terms of the

sample index?

- Draw the schematic
- Automatically generate a code that implements the systemDiscreteSystemImplementation[system, "implement"];
- Compute the successive output values{{y2}, {d2}} = implement[{1,10}, {d1}, {}];{{y3}, {d3}} = implement[{1,10}, {d2}, {}];
- Eliminate the initial states and find the relation between the output sampleseqns = Reduce[{y[n-1]==y2, y[n]==y3}, {d1}];
- Find the recurrence equationreducedEqn =(15 y[-1 + n] == -10 + 16 y[n]);
- Load knowledge for solving recurrence equations<<DiscreteMath`RSolve`
- Find a closed-form solutionRSolve[{reducedEqn,y[0]==0},y[n],n];

- Example: find the number of samples after which the output sequence reaches some value, say b
- Solve[y[n] == 10*b, n];
- Solution:
- Verification:

- 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}, ...}

Nonlinear

Power

Power

Gain

Scaled Signal

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

- Draw the schematic of algorithm
- Automatically generate a code that implements the systemDiscreteSystemImplementation[systemNR, "implementNR"];
- Compute the successive output values{{y2}, {d2}} = implementNR[{x,2}, {d1}, {}];{{y3}, {d3}} = implementNR[{x,2}, {d2}, {}];
- Eliminate the initial states and find the relation between the output sampleseqns = Reduce[{y[n - 1] == y2, y[n] == y3}, {d1}];
- Find the recurrence equationreducedEqn = (x*y[n-1]^2 == 2*y[n-1] - y[n]);
- Load knowledge for solving recurrence equations<<DiscreteMath`RSolve`
- Find a closed-form solutionsol = RSolve[{reducedEqn,y[0]==b},y[n],n];

- 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

- 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

- Programs provide knowledge about design and employ the knowledge in symbolic manipulation:
- automated generation of schematic objects and the corresponding implementation codes
- derivation of the transfer function, system properties and time response
- symbolic optimization

- Superiority of symbolic computation against numerical computation was shown by
- the example system with an algebraic loop; CAS yielded the exact solution while the traditional numeric approach failed
- the closed-form solution of a nonlinear LMS subsystem
- deriving the analytic expression for the error of the Newton-Raphson iteration

- Benefits of symbolic methods were highlighted from the viewpoint of
- Academia (derivation of time and frequency response, proving system properties)
- Industry (QAM, Hilbert transformer, LMS algorithm, verification of realizations, design alternatives in multirate systems)

http://library.wolfram.com/infocenter/TechNotes/4814/

http://www.schematicsolver.com

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