Chapter 12
Z transforms and DT LTI systems
The Z-transform is the analogue of the Laplace transform for discrete-time signals. It generalises the discrete-time Fourier transform to signals that are not absolutely summable. By taking the Z-transform of the impulse response of a DT LTI system, we obtain a generalization of the frequency response that is valid for systems that are not necessarily BIBO stable. This is called the transfer function.
In this topic we introduce the Z-transform of discrete-time signals, and the transfer function of a discrete-time LTI systems. The presentation is fairly brief since there are strong similarities between the Laplace transform and the Z-transform.
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12.1 The Z-transform
Recall that we motivated the Laplace transform as the Fourier transform Xσ(ω) of the rescaled signal xσ(t) = x(t)e−σt, thought of as a function of both σ and ω. We can motivate the Z- transform in a similar way. If x is a discrete-time signal, for any r > 0 we can define a rescaled signal xr[n] = x[n]r−n for all n. If we choose r large enough, we can ensure that the signal is absolutely summable, and so has a DTFT. We can then consider of the DTFT of the rescaled signal xr, i.e.,
Xr(ω) = x[n]r−ne−jωn = x[n](rejω)−n.
We can think of Xr(ω), the DTFT of the rescaled signal, as a function of a single complex
number z = rejω. This leads to the Z-transform.
Z-transform
If x is a DT signal, the Z-transform of x is the function Xˆ defined by
Xˆ(z) = x[n]z−n for all z ∈ RoC(x)
RoC(x)= z∈C: |x[n]z−n|<∞ n=−∞
is the region of convergence.
In words, the region of convergence consists of the set of complex numbers z for which the signal defined by x[n]z−n for all n, is absolutely summable. The following gives a slightly different alternative statement.
Region of convergence
The region of convergence RoC(x) is
the set of complex numbers z such that |z|−nx[n] is absolutely summable.
This holds because |x[n]z−n| = |x[n]||z|−n.
It follows directly that we can decide whether or not x is absolutely summable by examining
the region of convergence RoC(x).
This allows us to show that the discrete-time Fourier transform is just the Z-transform evaluated along the unit circle.
We now present the most important example of a Z-transform.
Region of convergence and absolute summability
A DT signal x is absolutely summable if and only if RoC(x) contains the unit circle.
Z-transform and discrete-time Fourier transform
If x is an absolutely integrable DT signal then RoC(x) contains the unit circle and the DT Fourier transform X and the Z-transform Xˆ of x are related by
X(ω) = Xˆ(ejω) for all ω.
Example 12.1: Z-transform of right-sided general complex exponential
Let α be a complex number and let x[n] = αnu[n] be a right-sided general complex expo- nential. The Z-transform Xˆ of x is
∞∞1z Xˆ(z)= αnu[n]z−n =(αz−1)n = 1−αz−1 = z−α
as long as |αz−1| < 1, or equivalently |z| > |α|. We have found that the region of conver-
which is everything outside a disk of radius |α|.
RoC(x)={z∈C : |z|>|α|}
Just as different continuous-time signals can have the same Laplace transforms, but are distinguished by the region of convergence, the same holds for discrete-time signals and the Z-transform.
Example 12.2: Z-transform of a left-sided general complex exponential
For comparison with Example 12.1, consider x[n] = −αnu[−n − 1] where α is a complex number. This is a left-sided signal and has Z-transform
−1 ∞ z/α z Xˆ(z)=− αnz−n =−(z/α)m =−1−(z/α) = z−α.
This time, however, the region of convergence is RoC(x)={z∈C : |z|<|α|}
which is the open disk of radius |α|.
12.1.1 Rational Z-transforms
Signals with rational Z-transforms arise very often. These are Z-transforms that have the form F(z) = B(z). It is often convenient to think of A and B as polynomials in z−1, and write
rational Z-transforms in the form
F(z)= b0 +b1z−1 +···+bmz−m a0 +a1z−1 +···+anz−n
b zm +bm−1 +···+b =zn−m 0 z m.
a0zn +a1zn−1 +···+an
All of the terminology related to rational functions (poles, zeros, etc) is also used to describe rational Z-transforms.
Example 12.3
Let x[n] = (1/4)nu[n] + (1/2)n cos(πn/2)u[n]. Using Euler’s formula we can write this as x[n] = (1/4)nu[n] + 0.5(1/2)nejπn/2u[n] + 0.5(1/2)ne−jπn/2u[n].
Simplifying gives
x[n] = (1/4)nu[n] + 0.5(j/2)nu[n] + 0.5(−j/2)nu[n]. Then, using Example 12.1 we see that
Xˆ(z)= z +0.5z+0.5z. z − 1/4 z − j/2 z + j/2
For the Z-transform to converge, it must converge for all of the three terms. This means that the region of convergence is the intersection of {z ∈ C : |z| > 1/4} and {z ∈ C : |z| > 1/2}, which is {z ∈ C : |z| > 1/2}. In this case the Z-transform is a rational function. To see this we can make a common denominator of (z − 1/4)(z + j/2)(z − j/2) = (z − 1/4)(z2 + 1/4) and simplify to obtain
ˆ 2z2 − (1/4)z + (1/4) 2 − (1/4)z−1 + (1/4)z−2 X(z)=z(z−1/4)(z2+1/4) =1−(1/4)z−1+(1/4)z−1−1/16.
Summary of Section 12.1
The Z-transform is a generalisation of the discrete-time Fourier transform that is valid for signals that are not absolutely summable. Two different discrete-time signals may have the same algebraic expression for their Z-transform, but will have different regions of convergence. Very often we are interested in signals that have Z-transforms that are rational functions (ratios of polynomials). In this case the roots of the numerator are called zeros, and the roots of the denominator are called poles.
12.2 The region of convergence for Z-transforms
The region of convergence of a Laplace transform was either a half-plane to the left of the left-most pole, a vertical strip between two poles, or a half-plane to the right of the right-most pole. For the Z-transform, the possible regions of convergence are essentially
• the entire complex plane
• the complement of a closed disk centered at the origin.
• an open anulus (i.e., a ring-shaped region between two concentric circles, not including the boundary) centered at the origin, or
• an open (i.e., not including the boundary) disk centered at the origin
Which of these occurs depends on whether the signal is left-sided, right-sided, or two-sided. (For discrete-time signals, the definitions of being left-, right-, and two-sided are the obvious analogues of the definitions for continuous-time signals.)
In addition we say that a signal x is
• causal ifx[n]=0whenevern<0and • anti-causal if x[n] = 0 whenever n > 0.
(This terminology makes sense because a causal signal is the impulse response of a causal LTI system.)
Region of convergence for left-, right-, and two-sided signals
• If a DT signal x is right-sided and causal then the region of convergence is the complement of a disk, i.e.,
RoC(x) = {z ∈ C : |z| > a}
for some real number a ≥ 0.
• If a DT signal x is left-sided and anti-causal then the region of convergence is a disk,
for some real number a ≥ 0.
RoC(x) = {z ∈ C : |z| < a}
• If a DT signal x is two-sided then the region of convergence is either empty or a ring
between two concentric circles, i.e.,
RoC(x) = {z ∈ C : b < |z| < a}
for some real numbers b < a.
(The reason this is a little more complicated than the Laplace transform case is that being causal or anti-causal affects whether ∞ or 0 are in the region of convergence of the Z-transform.)
We now summarize how the poles of a rational Z-transform are related to the region of convergence.
Region of convergence for rational Z-transforms
Suppose that x is a DT signal and the Z-transform Xˆ of x is a rational function. Then
• RoC(x) does not contain any poles of Xˆ
• if x is right-sided then the region of convergence is
RoC(x) = {z ∈ C : |z| > |p|} where p is the pole of Xˆ with the largest magnitude.
• if x is left-sided then the region of convergence is RoC(x) = {z ∈ C : |z| < |p|}
where p is the pole of Xˆ with the smallest magnitude.
• if x is two-sided then the region of convergence is
RoC(x) = {z ∈ C : |pi| < |z| < |pj|} for some pair of poles pi and pj of Xˆ.
Example 12.4
Consider the signal x[n] = (1/4)nu[n] + (1/2)n cos(πn/2)u[n] from Example 12.3. This has a rational Z-transform, and it is a right-sided signal. The poles are 1/4 and ±j/2. The largest magnitude poles are ±j/2 which have magnitude 1/2. The region of convergence of the Z-transform is then
RoC(x) = {z ∈ C : |z| > 1/2}.
Example 12.5
Consider the signal x(t) = (1/2)nu[n] − 2nu[−n − 1]. Note that this is a two-sided signal. By using the general results in Examples 12.1 and 12.2 we know that
Xˆ(z) = z + z = z z − (5/2) . z − 1/2 z − 2 (z − 1/2)(z − 2)
This Z-transform is rational and has poles at z = 1/2 and z = 2. We expect the region of convergence to be a an anulus containing no poles. Indeed the region of convergence must be
RoC(x) = {z ∈ C : 1/2 < |z| < 2}.
Note that the region of convergence contains the unit axis. This is consistent with the fact that x is absolutely summable.
Summary of Section 12.2
There are relationships between properties of a signal and the region of convergence of its Z-transform. Whether a signal is left-sided or right-sided or two-sided is reflected in the region of convergence. Similarly, a signal is absolutely summable if its region of convergence
contains the unit circle. For signals with rational Z-transforms, the region of convergence is closely related to the locations of the poles of the Z-transform.
12.3 Properties of the Z-transform
If we make transformations to a signal, then the Z-transform of the signal often transforms in a fairly simple way. In this section, we describe some of these properties of the Z-transform.
The most important of these is the convolution property. This states that convolution of two signals corresponds to multiplication of their Z-transforms. This greatly simplifies the task of finding the response of LTI systems to a given input.
Modulation
If x is a DT signal with Z-transform Xˆ and y[n] = z0nx[n] for all n (where z0 is a complex
number) then Furthermore
Yˆ(z) = Xˆ(z/z0).
RoC(y) = {z ∈ C : z/z0 ∈ RoC(x)}.
This is true because
Yˆ (z) = x[n]z0nz−n = x[n](z/z0)−n = Xˆ (z/z0).
This follows directly from the linearity of sums.
∞∞ n=−∞ n=−∞
Suppose x1 and x2 are DT signals with Z-transforms Xˆ1 and Xˆ2 and α, β ∈ C are complex numbers. If y[n] = αx1[n] + βx2[n] for all n then
Yˆ (z) = αXˆ1(z) + βXˆ2(z). The regions of convergence are related by
RoC(y) ⊇ RoC(x1) ∩ RoC(x2).
Convolution
Suppose x1 and s2 are DT signals with Z-transforms Xˆ1 and Xˆ2. If y[n] = (x1 ∗ x2)[n] is the convolution of x1 and x2 then
Yˆ (z) = Xˆ1(z)Xˆ2(z). The regions of convergence are related by
RoC(y) ⊇ RoC(x1) ∩ RoC(x2).
To see why this is true, we use the definition of the Z-transform and of convolution:
∞∞ Yˆ(z)= x1[k]x2[n−k] z−n.
We then exchange the order of summation to obtain
Yˆ(z)= x1[k]x2[n−k]z−n.
Since x1[k] is constant with respect to the inner sum, and z−n = z−(n−k)z−k we have ∞∞
Yˆ (z) = x1[k]z−k x2[n − k]z−(n−k). k=−∞ n=−∞
Making the change of variables n′ = n − k in the inner sum gives ∞∞
Yˆ (z) = x1[k]z−k x2[n′]z−n′ . k=−∞ n′ =−∞
Finally we can simplify this to obtain
Yˆ (z) = Xˆ1(z)Xˆ2(z)
If z ∈ RoC(x1) and z ∈ RoC(x2) then both sums will converge. It is possible for the region of convergence to be larger, because poles in Xˆ1 could be canceled by zeros in Xˆ2.
To see why this is true, we begin with the definition of the Z-transform
Yˆ (z) = x[n]∗z−n = x[n](z∗)−n∗
n=−∞ n=−∞ where we have used the fact that
z−n = (rejθ)−n = r−ne−jθn = [(re−jθ)−n]∗ = [(z∗)−n]∗. for all complex numbers z and all integers n. Then
Yˆ (z) = x[n](z∗)−n∗ = x[n](z∗)−n = Xˆ (z∗)∗ .
Conjugation
If x is a DT signal with Z-transform Xˆ and y[n] = x[n]∗ is the complex conjugate of x
and RoC(y) = RoC(x).
Yˆ ( z ) = Xˆ ( z ∗ ) ∗
Summary of Section 12.3
Various operations on signals give rise to corresponding operations on the Z-transforms of those signals. Because the Z-transform generalizes the discrete-time Fourier transform, these properties are very closely related to the corresponding properties of the Fourier transform. By thinking of a signal as being built out of ‘simple’ signals, we can use these properties to compute the Z-transform of a complicated signal from knowledge of the Z-
transforms of a few ‘simple’ signals.
12.4 Z-transforms and DT LTI systems
The Z-transform is a frequency domain-like representation of discrete-time signals, even if they are not absolutely summable. By taking the Z-transform of the impulse response of an LTI system, we obtain a generalization of the frequency response of a DT LTI system that is valid even if the system is not BIBO stable. This representation is called the transfer function of an LTI system.
In this topic we breifly summarize the properties of the transfer function for DT LTI systems, and show how to use the Z-transform and inverse Z-transform to find the response of a DT LTI system to a given input.
12.5 Transfer functions of DT LTI systems
The transfer function of a DT LTI system is related to the system output when the input is a general complex exponential x[n] = zn.
Response of LTI system to general complex exponential
Let z be a complex number and let x[n] = zn be a general complex exponential signal. If an LTI system has impulse response h and input x then there is a function Hˆ such that
the output is
We call the complex-valued function Hˆ the transfer function of the system.
y[n] = (h ∗ x)[n] = Hˆ (s)zn.
To see why this is true we use the convolution formula:
y[n] = h[k]zn−k
=zn h[k]z−k k=−∞
= Hˆ ( z ) z n .
Notice that Hˆ is precisely the Z-transform of the impulse response.
The frequency response is the DTFT of the impulse response of a BIBO stable LTI system. The transfer function is the Z-transform of the impulse response of an LTI system. We can use the relationship between the Fourier transform and the Z-transform to relate the frequency response and the transfer function.
Transfer function and impulse response
The transfer function of a DT LTI system is the Z-transform of the impulse response of the system.
Transfer function and frequency response
If Hˆ is the transfer function of a BIBO stable DT LTI system, and H is the frequency
response of the system then
H ( ω ) = Hˆ ( e j ω ) .
Relating the input, output, and transfer function of LTI systems
uppose an LTI system has transfer function Hˆ . If Xˆ is the Z-transform of the input of the system and Yˆ is the Z-transform of the output of the system then
Yˆ (z) = Hˆ (z)Xˆ (z).
One way to see why this holds is to use the convolution property of Z-transforms. Clearly the formulas for transfer functions of interconnections of systems we found for the continuous-time setting remain true in the discrete-time setting.
• If two LTI systems are connected in series, then the transfer function is the product of the transfer functions of the individual systems.
• If two LTI systems are connected in parallel, then the transfer function is the sum of the transfer function of the individual systems.
• If a system is a feedback interconnection of two LTI systems S1 and S2 shown below
then the transfer function of the overall system is
Hˆ ( z ) = Hˆ 1 ( z ) .
1 − Hˆ1(z)Hˆ2(z)
Summary of Section 12.5
If the input to a DT-time LTI system is zn (where z is a complex number), the output is Hˆ (z)zn. The function Hˆ is called the transfer function of the system. The transfer function is the Z-transform of the impulse response of the system. For BIBO stable systems, the transfer function and the frequency response are related by H(ω) = Hˆ(ejω). If an LTI system has transfer function Hˆ and input with Z-transform Xˆ then the output has Z- transform Yˆ (z) = Hˆ (z)Xˆ (z). Finally, when we interconnect subsystems in series, parallel, or feedback, the transfer function of the overall system is related to the transfer function
of the subsystems in exactly the same way in CT and DT.
12.6 The region of convergence and properties of LTI systems
We have seen that there are close connections between the structure of the region of convergence of a Z-transform and whether the associated signal is left- or right- or two-sided, and whether it is absolutely summable. For LTI systems, there are close connections between whether a system is causal or BIBO stable, and the region of convergence of the Z-transform.
Region of convergence and causal systems
If a DT LTI system is causal then its impulse response is right-sided. It follows that
• for a causal DT LTI system the region of convergence of the transfer function has the form
{z ∈ C : |z| > a}
• for a causal DT LTI system with a rational transfer function the region of convergence
for some real number a. is
{z ∈ C : |z| > |p|} where p is the rightmost pole of the transfer function.
Region of convergence and BIBO stability
A DT LTI system is BIBO stable if and only if its impulse response if absolutely summable. It follows that a DT LTI system is BIBO stable if and only if the region of convergence of its transfer function contains the imaginary axis.
Region of convergence for causal BIBO stable systems
• A causal DT LTI system is BIBO stable if and only if the region of convergence of its transfer function is of the form
{z ∈ C : |z| > a}
where a < 1.
• A causal LTI system with rational transfer function is BIBO stable if and only if all of its poles are have magnitude less than one.
Summary of Section 12.6
The region of convergence of the transfer function is closely related to properties of the cor- responding DT LTI system. For instance, the region of convergence contains the unit circle if and only if the system is BIBO stable, and the region of convergence is the complement of a disk if and only if the system is causal.
12.7 Inverse Z-transforms and computing system response
We have two options to compute the system response for general (not necessarily absolutely summable) input signals and general (not necessarily BIBO stable) LTI systems. The first is based on convolution:
The second is based on exploiting the convolution property of the Z-transform
Time-domain approach
1. Find the impulse response.
2. Compute the convolution of the impulse response and the input.
Z-transform-based approach
1. Find the transfer function Hˆ.
2. Find the Z-transform Xˆ of the input.
3. Compute the -ztransform Yˆ (z) = Hˆ (z)Xˆ (z) of the output. 4. Find the signal y with Z-transform Yˆ(z).
The last step of the Z-transform-based approach, is the only one we haven’t discussed already. Given a Z-transform and its region of convergence, how can we find the signal with that Z- transform? This process is called finding the inverse Z-transform. There is a formula for the inverse Z-transform. It is based on integration in the complex plane, and is not so useful in practice. In practice, we usually compute inverse Z-transforms by decomposing a Z-transform as a sum of simple pieces, ‘looking up’ the inverse Z-transforms of the pieces in a table (possibly also using other Z-transform properties), and then using linearity.
This approach is particularly straightforward when the Z-transform is rational. Then we take the same basic approach as for computing inverse Laplace transforms. We decompose the Z-transform into partial fractions, and then compute the inverse Z-transform term-by-term, from the tables.
Summary of Section 12.7
We can compute the output of a discrete-time LTI system for a given input using Z- transforms. We compute the Z-transform Xˆ of the input, then obtain the Z-transform Yˆ of the output using Yˆ(z) = Hˆ(z)Xˆ(z). Finally, we compute the inverse Z-transform of Yˆ, that is, the continuous-time signal with Z-transform Yˆ and the correct region of convergence. If Yˆ is a rational function, we do this by decomposing Yˆ into partial fractions, and then finding the inverse Z-transform of each te
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