Cosc 3451 Signals and Systems - York University · 2005-09-13 · 1 Cosc 3451 Signals and Systems...

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Cosc 3451 Signals and Systems

Systems Terminology and Propertiesof Systems

What is a system?

• an entity that manipulates one or more signals toyield new signals (often to accomplish a function)

• can be thought of as an interconnection ofcomponents, devices and subsystems (operations)

Figure 1.2 (p. 3)Elements of a communication system. The transmitter changes the messagesignal into a form suitable for transmission over the channel. The receiver

processes the channel output (i.e., the received signal) to produce an estimate ofthe message signal.

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Signals and Systems, 2/E by Simon Haykin and Barry Van VeenCopyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 1.3 (p. 5)(a) Snapshot of Pathfinder exploring thesurface of Mars. (b) The 70-meter (230-foot) diameter antenna located atCanberra, Australia. The surface of the70-meter reflector must remain accuratewithin a fraction of the signal’swavelength. (Courtesy of Jet PropulsionLaboratory.)


Figure 1.4 (p. 7)Block diagram of a feedback control system. The controller drives theplant, whose disturbed output drives the sensor(s). The resultingfeedback signal is subtracted from the reference input to produce an errorsignal e(t), which, in turn, drives the controller. The feedback loop isthereby closed.

• Where H{} is a function describing theoverall action of the system

Continuous-time Systemx(t) y(t)

Discrete-time Systemx[n] y[n]

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Systems can be connected in:H1In OutH2Series:

H1In Out

H2

Parallel: +

H1In Out

H2

Feedback: +

and various other combinations


Figure 2.32 (p. 162)Symbols for elementary operations in block diagram descriptions ofsystems. (a) Scalar multiplication. (b) Addition. (c) Integration forcontinuous-time systems and time shifting for discrete-time systems.


Figure 1.50 (p. 54)Discrete-time-shift operator Sk, operating on the discrete-time signal x[n]to produce x[n – k].

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Figure 1.51 (p. 54)Two different (but equivalent) implementations of the moving-average system:(a) cascade form of implementation and (b) parallel form of implementation.

Properties• Memory

– if system output y[n] {or y(t)} depend onprevious values of system input x[n] or outputy[n] it exhibits memory, i.e.

– if y[n] depends only on x[n] it is memoryless

• Causality– in a causal system, outputs depend only on

current and previous inputs, e.g.• y[n] = x[n-2] (delay)

– if output of system depends on future values itis not causal, e.g.

• y[n] = x[n+2] (advance)

– physical systems/real-time systems are causal

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• Stability– a stable system gives a bounded output, y[n],

for a bounded input, x[n]

Tacoma narrows bridge

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Figure 1.69 (p. 79)Block diagram of first-order recursive discrete-time filter. The operator Sshifts the output signal y[n] by one sampling interval, producing y[n – 1].The feedback coefficient p determines the stability of the filter.

• Time/shift invariance– behaviour of a time-invariant (shift-invariant)

system is fixed over time– doesn’t burn in, age, learn, adapt– let y1[n] be the output of system to x1[n] then

for shift invariance,• the output y2[n] to input x2[n] = x1[n-n0] must bey2[n] = y1[n-n0] for all n0 and x1[n]


Figure 1.55 (p. 61)The notion of time invariance. (a) Time-shift operator St0 precedingoperator H. (b) Time-shift operator St0 following operator H. These twosituations are equivalent, provided that H is time invariant.

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Figure 1.68 (p. 78)Tapped-delay-line model of a linear communication channel, assumed tobe time-invariant.

• Linearity– A linear system satisfies the principle of

superposition, that is

– linearity is a very important and useful property– never met in physical systems but useful

approximation– superposition property means that response to

a complex input is the superposition ofresponses to components of the input, this willprove very useful

– if input is always zero, then output will alwaysbe zero. Useful in justifying the assumption ofconditions of initial rest in LTI systems.

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Figure 1.56 (p. 64)The linearity property of a system. (a) The combined operation ofamplitude scaling and summation precedes the operator H for multipleinputs. (b) The operator H precedes amplitude scaling for each input; theresulting outputs are summed to produce the overall output y(t). If thesetwo configurations produce the same output y(t), the operator H is linear.


Figure 1.53 (p. 59)Series RC circuit driven from an ideal voltage source v1(t), producingoutput voltage v2(t).


Figure 1.64 (p. 73)Mechanical lumped model of an accelerometer.

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Figure 1.54 (p. 59)The notion of system invertibility. The second operator Hinv is the inverseof the first operator H. Hence, the input x(t) is passed through the cascadecorrection of H and H-1 completely unchanged.

LTI Systems• a very important class of systems are the

Linear Time-Invariant (LTI) systems• linearity and sampling/sifting properties of

the pulse/impulse give us the convolutionsum and integral representations ofresponse of systems– sampling property

• superposition:– signal can be represented as a combination of a

number of component signals, i.e.shifted/delayed pulses or impulses

– in LTI system overall output is superpositionof responses to the component signals

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• Sifting sum (discrete):

Σ

• Sifting integral (continuous):

• integral is ‘sum’ of impulse samples of x(t)placed infinitely close together (fig 2.12)

Convolution Sum and Integral

• Discrete (Convolution Sum):– input can be represented as a sum (sifting

sum)of appropriately shifted and scaled pulses– in LTI system, overall output is superposition

of responses to these pulses– if response to input is H{x[n]} then response to

each pulse in the sifting sum is shifted andscaled version of h[n] = H{δ[n]}

– example (on board)

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– if system is time-invariant, h[n] is constant forimpulse at any n

H{}x[n] y[n]

– for each y[n] need to evaluate a set of signals,vk[n] = x[k]h[n-k] and sum over k

OR

– consider h[n-k] = h[-(k-n)], reflected and time-shifted version (by –n) of h[k]

– evaluate one signal at each interval,wn[k]=x[k]h[n-k] and sum it up over k

h[k]

h[-k]

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Figure 2.1 (p. 99)Graphical example illustrating therepresentation of a signal x[n] as aweighted sum of time-shiftedimpulses.


Figure 2.2a (p. 100)Illustration of the convolution sum. (a) LTI system with impulse responseh[n] and input x[n].

• commutativity:– h[n]* x[n]= x[n]* h[n]

• distributivity:– h[n]*(x[n] + z[n]) = h[n]* x[n] + h[n]* z[n]

• associativity:– h[n]* (x[n]* z[n]) = ( h[n]* x[n] )* z[n]

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• Continuous (Convolution integral)– directly analogous to discrete case

– each impulse in sifting integral drives animpulse response; like discrete case butinfinitely close together

– manual method1. Graph x(τ), h(τ) as a function of τ. To get

function h(t-τ) reflect h(τ) about τ=0 andtime shift-t

2. begin with t large, negative3. obtain functional form of x(τ) h(t-τ) = wt(τ).

This is often constant for a range of t.4. Integrate wt(τ) from τ = -∞ to ∞ to get y[n]

for each time texamples…

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Properties of LTI Systems andConvolution

• Parallel System

h1[n]x[n]

h2[n]+

y1[n]

y2[n]y[n]

h1(t)x(t)

h2(t)+

y1(t)

y2(t)

y(t)

• Series System x[n] y[n]h1[n] h2[n]h12[n]

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Figure 2.21 (p. 131)(a) Reduction of parallel combination of LTI systems in upper branch of Fig. 2.20.(b) Reduction of cascade of systems in upper branch of Fig. 2.21(a).(c) Reduction of parallel combination of systems in Fig. 2.21(b) to obtain an equivalent system for Fig. 2.20.

• Memoryless

– but memoryless system can only depend onx[n] and cannot depend on x[n-k] for k≠0

• Causality– causal system cannot respond to input before it

occurs– y[n] cannot depend on x[n-k] for (n-k) > n, i.e.

for k <0

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• BIBO Stability

• Step response

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• Response to sinusoid/complex exponential(preview of Fourier Transform)

– H(jω) is a function of ω not t -> frequencyresponse

– in general it is complex

amplituderesponse

phaseresponse

– input is sinusoid/periodic exponential offrequency, ω.

– Output is same frequency sinusoid withmagnitude |H(jω)|*Ax and phase equal to inputphase + arg(H(jω))

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Bode plot

• for discrete, we only need to plot from -π toπ

Where do h[n], h(t) come from?

• until now we’ve assumed they are known• LTI systems can be described by linear

constant coefficient difference/differentialequations

• h(t) or h[n] is the solution of theseLCCDEs to a impulse/pulse forcingfunctionsolving LCCDE with appropriate initial

conditions and impulse/pulse input giveh(t)/h[n]

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Continuous-time impulseresponse

• recall: N-th order LCCDE

• to solve for y(t) need two sub-solutions:– the forced response: a ‘particular’ solution

which depends on the input x(t)– the natural response: a ‘homogeneous’

solution to

• differential equation on its own is notsufficient; we need to define auxiliary(initial) conditions to determine theresponse

• Usually assume initial rest, that is is x(t) =0 for t <= t0 then

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• causal LTI systems imply this condition –no stored energy/memory when no inputhas been applied

• Thus, causal LTI systems (an importantclass of real world systems) can bedescribed by a LCCDE under conditions ofinitial rest

• for an LTI system the forced response(particular solution) is linear with respect tothe input

• also, natural response is linear with respectto the initial conditions but completeresponse is not

• therefore, for time invariance we needconditions of initial rest so that output doesnot depend on timing of input except for atime shift

• so for h(t), we need to solve LCCDE forimpulse input (note that impulse responseis a function of the terms in naturalresponse)– see 2.56 for one way– another way is to find s(t), the step response or

the solution of the LCCDE with u(t) as input

– use transform techniques

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Discrete-time pulse response• N-th order Linear Constant Coefficeint

Difference Equations

• need to know y[-1] .. y[-N] as initialconditions to calculate y[0]

• if x[n] = 0 for n<n0 then conditions ofinitial rest are y[n0-N] = y[n0-N+1]=…=y[n0-1]=0

• LCCDE with conditions of initial rest givecasual LTI systems

• IIR system– y[n] depends on previous values of y

recursively– impulse response can be infinite in length

(infinite impulse response or IIR) since effectsof input values before x[n-M] are stored

• FIR system– if N = 0

– y[n] can only be influenced by the previous Mvalues of x – finite impulse response (FIR)

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• Solution of h[n] (a function of terms innatural response):– find s[n], h[n] = s[n]-s[n-1]– algebraically rearrange for y[n] in terms of

x[n], x[n-1], … and y[n], y[n-1], … and solvefor x[n] = δ[n] and initial rest conditions, e.g.

– closed form solution (e.g. problem 2.55)– for FIR (or truncated IIR) apply δ[n] and solve

for h[n] only over the M intervals– use transform methods


Figure 2.37 (p. 166)Block diagram representations of a continuous-time LTI system describedby a second-order integral equation. (a) Direct form I. (b) Direct form II.

Cosc 3451 Signals and Systems - York University · 2005-09-13 · 1 Cosc 3451 Signals and Systems...

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