SIG Compendium 01 Part2

Cologne University of Applied Sciences SIG Compendium Prof. Dr. Rainer Bartz

4 ANALYSIS OF DT-SYSTEMS IN THE TIME-DOMAIN311312

The most frequent task in systems analysis:313 given: a SISO system (represented e.g. by its system equation), and an input signal x[nT].314 to be determined: the systems response (its output signal) y[nT].315Variety of methods:316 solving difference equations Mathematics.317 recursive numerical method chapter 4.1318 convolution chapter 4.2 + 4.3319 z-transform chapter 5320

321322

4.1 The Recursive Numerical Method (C11.2)323This method assumes that the system is causal so that its output at a given point in time only depends on present or past signal 324samples. It requires that the system equation is available.325

326A general representation for system equations of causal LTI systems is:327

l

0ii

k

1ii i)T]-x[(n(bi)T]-y[(n(-ay[nT] ))328

From this representation the following scheme may be set up and used to subsequently determine the output samples of y[nT]:329330331

input output b0 b1 b2 ... bl -ak ... -a2 -a1

n x[nT] x[(n-1)T] x[(n-2)T] ... x[(n-l)T] y[(n-k)T] ... y[(n-2)T] y[(n-1)T] y[nT]n0 ... ... ... ... ... ...(*) ... (*) ... (*) ... (*) ...n0+1 ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ...

332

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Each row is used to calculate y[nT] from the other entries in the row and the coefficients in the header row according to:3331)T]-y[(na-...-k)T]-y[(na-l)T]-x[(nb...1)T]-x[(nbx[nT]by[nT] 1kl10 334

335For the algorithm to work it is required to336a) determine a starting point n0; this is usually the left border of the input signal and may in many cases be set to n0=0.337b) know about the initial conditions. Initial conditions are the samples that occurred at the output signal before the set starting point. 338

They are y[(n0-1)T], y[(n0-2)T], y[(n0-k)T], marked in the scheme by (*). Thus a system of order k requires k initial conditions. In 339many cases all initial conditions are 0.340

341Such algorithm is well structured for hardware implementation, requiring multiply-add units and shift operations.342A drawback of this algorithm is that it only produces numerical values. A mathematical expression for y[nT] is often hard to determine 343from them.344

345346347

4.2 Unit Pulse Response (C11.3)348The unit pulse response of a system is the output signal when the input signal is a unit pulse function G[nT]; its symbol is h[nT].349It is unique for a given system and contains all information available on the system itself!350

351The unit pulse response of a system may be determined 352 with mathematical methods, or353 potentially by applying the recursive numerical method, or354 by use of the z-transform ( chapter 5; preferred method), or355 estimated through measurements of a pulse or step response.356

357A system is causal if and only if h[nT]=0 for all n


4.3 Superposition Concept, Convolution, and some Properties (C11.4+5)362Given363 an LTI system, uniquely specified by its unit pulse response h[nT], and 364 an input signal x[nT].365

366It can be shown that the output signal y[nT] can be calculated through the following sum:367

ff

m

m)T]-h[(nx[mT]y[nT] ;368

Such operation is called convolution, with a short-form notation: y[nT]=x[nT]

h[nT] .369370371372

Characteristics of the convolution operation:373 convolution is commutative: x[nT] h[nT] = h[nT] x[nT]374 convolution is associative: (x[nT] h[nT]) g[nT] = x[nT] (h[nT] g[nT])375 convolution is distributive (over addition): x[nT] (h[nT] + g[nT]) = x[nT] h[nT] + x[nT] g[nT]376 if x[nT] and h[nT] show finite extent, left and right border usually add 377

during convolution, and the extent calculates from nye = nxe + nhe - 1.378 the unit pulse function is the neutral element of convolution: x[nT] G[nT] = x[nT]379 convoluting a signal with a shifted unit pulse simply shifts the signal: x[nT] G[(t-t0)T] = x[(t-t0)T]380

381382383

The following is a detailed step-by-step approach to solve this convolution sum and to determine the system response in the time-384domain; to ease the procedure it is usually performed considering only the index values and omitting the time axis. Note that with 385increased experience one will not need to perform each individual step.3861) Sketch a graph of the input signal x[n] and the system impulse response h[n].387

Write down a functional expression for x[n] and h[n].3882) Sketch a graph of x[m], h[m] and h[-m]; these are based on m as independent variable.389

Sketch a graph of h[n-m] and denote its significant points.390Write down a functional expression for x[m] and h[n-m].391

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3) For each value of m in the range of -f..f (usually interval-wise):392 Sketch a graph of the relationship between x[m] and h[n-m] over m.393 Multiply x[m] and h[n-m] for each point on the m-axis.394 Build the sum over that product (eventually piecewise): f

f

mm]-h[nx[m]y[n] .395

4) Finally combine the partial results into a piecewise specification for y[n].396397

Generally the partial sums to be determined in step 3) are infinite sums and it may be difficult to specify a resulting expression; only 398approximations might be available in such case.399

400401402

Special cases:403

if the system is causal: f

n

mm]-h[nx[m]y[n]404

if input signal x[nT] and unit pulse response h[nT] both are finite with extents nxe, nhe, convolution can be performed in (nxe+nhe-1) 405single steps, each one producing a single sample value for y[nT], and the remaining samples of y[nT] being 0.406

407408

It may be worth noticing that the sum of the sample indices of x[m] and h[n-m] always is n.409410411412413414415416417418

419

9aSIG_Compendium_01.doc / 24.03.14 / RBz SIG Compendium


5 THE Z-TRANSFORM (C14)420Determining the output of an LTI system through the convolution operation may soon become tedious if the system's complexity 421increases. Moreover, several important questions cannot be answered at all in the time-domain. 422This chapter introduces a new concept for signal and systems analysis purposes, with support for some system design decisions: the 423z- transform. It allows to determine LTI system responses in a systematic way (and more generally to ease the solution of linear 424difference equations), and it also provides a quantitative measure for system stability.425

426A prerequisite for all considerations in this chapter (except parts of section 5.4) is that signals are zero for all times nTrc.437

438One of the most important characteristics of the z-transform is that it allows to uniquely reconstruct x[nT] from X(z) using the inverse z-439transform. Thus, transform and inverse transform can be applied subsequently and always switch between the same two functions.440

The inverse z-transform is defined as

0n

0n

for

0

dzzX(z)j21

x(nT)ir

1-n

t

S

, with ri>ROC, though it is usually not evaluated directly.441

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442Notations:443 As z is often called a frequency in a wider sense, especially in the context of z-transforms, X(z) is then consequently called a signal 444

in the frequency-domain; also the term z-domain is widely used.445 The operation to produce X(z) from x[nT] is called z-transform and is sometimes denoted by X(z)=ZZ{x[nT]}; the operation to produce 446

x[nT] from X(z) is called inverse z-transform and is sometimes denoted by x[nT]=Z-1{X(z)}. 447 As the transform can be performed in both directions, a frequent notation is x[nT] X(z) with the open circle representing the 448

time-domain and the filled circle representing the frequency-domain.449 It has become common practice to specify signals in the time-domain with lowercase letters and signals in the frequency-domain 450

with uppercase letters.451452453454

Determining z-transform and inverse transform results:455There is a three-tier decision process to perform the transform of a time-domain signal x[nT] into X(z): 4561. Determine whether the signal x[nT] can be found in a table of z-transform pairs.4572. Determine whether any transform theorems can be applied to the signal x[nT] so that the results can be found in a table of z-458

transform pairs.4593. If neither of the first two actions leads to the desired transform, the definition sum for X(z) must be evaluated.460

461The inverse transform of X(z) into its time-domain signal x[nT] can be achieved by following decision hierarchy:4621. Determine whether the signal X(z) can be found in a table of z- transform pairs.4632. Determine whether any transform theorems can be applied to the signal X(z) so that the results can be found in a table of z-464

transform pairs.4653. Determine whether partial fraction expansion can be applied to X(z) so that the results can be found in a table of z-transform pairs466

or transform theorems can be applied.4674. If neither of the first three actions leads to the desired transform, the definition integral for the inverse transform must be evaluated.468Note, that for the inverse z-transform partial fraction expansion helps to avoid solving the integral in most cases.469

470471472473



5.2 Theorems of the z-Transform474The following list holds a set of important theorems; they can be seen as rules helping to determine the z-transform of a signal.475R1: Linearity476

Let x[nT] X(z) and y[nT] Y(z) a.x[nT]+b.y[nT] a.X(z) + b.Y(z) for a,b: const.477R2: Time shift478

Let x[nT] X(z) x[(n-m)T] X(z).z-m for mt0, m: const.479R3: z-scaling480

Let x[nT] X(z) )azX(x[nT]an for a: const.481

R4: Multiplying with (nT)k482

Let x[nT] X(z) ^ `x[nT](nT)dzdTznT]x(nT) 1kk Z[ for k>0, k: const.483

R5: Difference484Let x[nT] X(z) x[nT]-x[(n-1)T] (1-z-1).X(z)485

R6: Sum486

Let x[nT] X(z) 1-n

0i z-1X(z)x[iT]

487

R7: Convolution488Let x[nT] X(z) and y[nT] Y(z) x[nT] y[nT] X(z) . Y(z)489

R8: Limit Theorems490R9a: Initial Value491Let x[nT] X(z) (X(z))limx[0]

z fo 492R9b: Final Value493Let x[nT] X(z) X(z)z-(1limx[nT]lim)](x[ -1

1zn f ofo )494

only in case all poles of (1-z-1).X(z) lie inside the unit circle495496497498

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5.3 Inverse z-Transform499In many cases the z-transform X(z) of a signal x[nT] turns out to be a rational function of z, consisting of a numerator polynomial and a 500denominator polynomial. If x[nT]=0 for ns.521

522523524



5.3.2 partial fraction expansion525Rational functions with real coefficients as given above can be expanded to only contain non-negative exponents of z and then 526factorized in numerator and denominator, leading to their factor form:527

)-(z...)-(z)-(z...)-(z)-(zzCX(z)

m21

21rm-DDD

EE 528529

This representation shows the following important characteristics:530 The values Di are called the poles of X(z); the number m of poles of X(z) equals the order of the denominator polynomial. If m


A very important aspect of this method is that it also can be applied in cases where the system output y[nT] has not been 0 for all 584times nT0, m: const.588The sample values y[iT] are the initial conditions already encountered in section 4.1 above.589

590This allows investigating the system response on an input signal in situations where the output signal from a previous stimulus has not 591yet disappeared.592

593594595596

5.5 The Transfer Function of a System597The ratio Y(z)/X(z) is called the transfer function H(z) of a system. It is a unique specification of a system.598Hence Y(z)=H(z).X(z); the z-transform Y(z) of the output signal y[nT] of such system can be calculated from the transfer function H(z) 599and the Laplace transform X(z) of the input signal x[nT] of the system. As both, X(z) and H(z), are most likely rational functions, Y(z) 600will become a rational function as well, and partial fraction expansion is a systematic approach to determine y[nT] for a given x[nT] and 601a given system.602

603Besides being useful for calculating system responses, the transfer function provides information about the stability of a system and 604allows to determine the frequency characteristics of a system (see next sections).605

606Several options exist to determine the transfer function of a causal LTI system:607The transfer function H(z) can be determined from the system equation according to the previous section:608

zaza...zazaazbzb...zbzbbH(z) k-k

1)-(k-1-k

2-2

1-10

-ll

1)--(l1-l

-22

-110

609

As above, for real systems all coefficients ai and bi are real values.610611

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The transfer function H(z) can also be determined from the unit pulse response h[nT] of the system simply by H(z)=ZZ{h[nT]}.612613

Furthermore the transfer function can be determined graphically from any block diagram given for the system. For this purpose the 614complete block diagram is sketched in the z-domain (i.e. signals are denoted X(z), Y(z), instead of x[nT], y[nT], ) and each block 615is given a content that corresponds to its transfer function (i.e. a delay block will be given the content of z-1 instead of D, as z-1 is the 616transfer function of a delay block; constant values in a block remain unchanged).617Having done this, the following block operations may be performed to subsequently reduce the number of blocks (and thereby the618block diagram's complexity) until only one block is left. The content of that block is the transfer function H(z) of the system.619

620Block diagram operations (in the z-domain!):621 two blocks (with content H1(z) and H2(z)) in series can be replaced by one block with content H1(z).H2(z).622 two blocks (with content H1(z) and H2(z)) in parallel can be replaced by one block with content H1(z)+H2(z).623 two blocks (with content H1(z) in forward and H2(s) in feedback direction) in a basic feedback arrangement can be replaced by one 624

block with content (z)H(z)H1

(z)H21

1# . The sign is contrary to that of the feedback signal at the combination (sum point).625

A branch may be moved from before a block to behind that block and vice versa; a compensating block has to be added to the 626block diagram so that the overall functionality remains unchanged.627

A sum point may be moved from before a block to behind that block and vice versa; a compensating block has to be added to the 628block diagram so that the overall functionality remains unchanged.629

One should avoid swapping a branch and an adjacent sum point; the compensations needed to keep the overall functionality 630unchanged usually introduce additional complexity into the block diagram instead of simplifying it.631

632633634635636



5.6 Stability and Frequency Response of a System637

5.6.1 System Stability638The definition of the BIBO-stability as given before is not useful to determine whether or not a given system is stable. For this purpose 639a set of stability criteria have been developed. Only one of them shall be described here. For stability analysis the initial conditions are 640assumed to be 0.641

642Stability analysis based on the poles of the system643This provides complete information on a system's stability but requires that H(z) is given and all of its poles are exactly 644known/determined.645Given a causal LTI system with a rational transfer function H(z)=N(z)/D(z), where N(z) is the numerator polynomial and D(z) is the 646denominator polynomial.647

648The following procedure determines the stability characteristics of the system:6491. Determine all zeros Ei and polesDi of H(z) and specify H(z) in its factor form.6502. If N(z) and D(z) have a common factor (z-z0), both shall be divided by that factor, and only the resulting H(z) is investigated further.6513. The system is652

stable if all poles Di of H(z) are inside the unit circle (i.e. |Di|

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