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Introduction
This module will look at some of the basic properties of the Z-Transform (Section 9.2) (DTFT).
Note
We will be discussing these properties for aperiodic, discrete-time signals but understand that very similar properties hold for continuous-time signals and periodic signals as well.
Discussion of Z-Transform Properties
Linearity
The combined addition and scalar multiplication properties in the table above demonstrate the basic property of linearity. What you should see is that if one takes the Z-transform of a linear combination of signals then it will be the same as the linear combination of the Z-transforms of each of the individual signals. This is crucial when using a table (Section 8.3) of transforms to find the transform of a more complicated signal.
Example \(\PageIndex{1}\)
We will begin with the following signal:
\[x[n]=a f_{1}[n]+b f_{2}[n] \nonumber \]
Now, after we take the Fourier transform, shown in the equation below, notice that the linear combination of the terms is unaffected by the transform.
\[X(z)=a F_{1}(z)+b F_{2}(z)\nonumber \]
Symmetry
Symmetry is a property that can make life quite easy when solving problems involving Z-transforms. Basically what this property says is that since a rectangular function in time is a sinc function in frequency, then a sinc function in time will be a rectangular function in frequency. This is a direct result of the symmetry between the forward Z and the inverse Z transform. The only difference is the scaling by \(2\pi\) and a frequency reversal.
Time Scaling
This property deals with the effect on the frequency-domain representation of a signal if the time variable is altered. The most important concept to understand for the time scaling property is that signals that are narrow in time will be broad in frequency and vice versa. The simplest example of this is a delta function, a unit pulse with a very small duration, in time that becomes an infinite-length constant function in frequency.
The table above shows this idea for the general transformation from the time-domain to the frequency-domain of a signal. You should be able to easily notice that these equations show the relationship mentioned previously: if the time variable is increased then the frequency range will be decreased.
Time Shifting
Time shifting shows that a shift in time is equivalent to a linear phase shift in frequency. Since the frequency content depends only on the shape of a signal, which is unchanged in a time shift, then only the phase spectrum will be altered. This property is proven below:
Example \(\PageIndex{2}\)
We will begin by letting \(x[n]=f[n−\eta]\). Now let's take the z-transform with the previous expression substituted in for \(x[n]\).
\[X(z)=\sum_{n=-\infty}^{\infty} f[n-\eta] z^{-n} \nonumber \]
Now let's make a simple change of variables, where \(\sigma=n-\eta\). Through the calculations below, you can see that only the variable in the exponential are altered thus only changing the phase in the frequency domain.
\begin{aligned}
X(z) &=\sum_{n=-\infty}^{\infty} f[\sigma] z^{-(\sigma+\eta)} \\
&=z^{-\eta} \sum_{\sigma=-\infty}^{\infty} f[\sigma] z^{-\sigma} \\
&=z^{-\eta} F(z)
\end{aligned}
Convolution
Convolution is one of the big reasons for converting signals to the frequency domain, since convolution in time becomes multiplication in frequency. This property is also another excellent example of symmetry between time and frequency. It also shows that there may be little to gain by changing to the frequency domain when multiplication in time is involved.
We will introduce the convolution integral here, but if you have not seen this before or need to refresh your memory, then look at the discrete-time convolution (Section 4.3) module for a more in depth explanation and derivation.
\[\begin{align}
y[n] &=\left(f_{1}[n], f_{2}[n]\right) \nonumber \\
&=\sum_{\eta=-\infty}^{\infty} f_{1}[\eta] f_{2}[n-\eta]
\end{align} \nonumber \]
Time Differentiation
Since discrete LTI (Section 2.1) systems can be represented in terms of difference equations, it is apparent with this property that converting to the frequency domain may allow us to convert these complicated difference equations to simpler equations involving multiplication and addition.
Parseval's Relation
\[\sum_{n=-\infty}^{\infty} x[n] x *[n]=\int_{-\pi}^{\pi} F(z) F *(z) d z \nonumber \]
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.
Modulation (Frequency Shift)
Modulation is absolutely imperative to communications applications. Being able to shift a signal to a different frequency, allows us to take advantage of different parts of the electromagnetic spectrum is what allows us to transmit television, radio and other applications through the same space without significant interference.
The proof of the frequency shift property is very similar to that of the time shift; however, here we would use the inverse Fourier transform in place of the Fourier transform. Since we went through the steps in the previous, time-shift proof, below we will just show the initial and final step to this proof:
\[z(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega-\phi) e^{j \omega t} d \omega \nonumber \]
Now we would simply reduce this equation through another change of variables and simplify the terms. Then we will prove the property expressed in the table above:
\[z(t)=f(t) e^{j \phi t} \nonumber \]
Properties Demonstration
An interactive example demonstration of the properties is included below:
Summary Table
| Property | Signal | Z-Transform | Region of Convergence |
|---|---|---|---|
| Linearity | \(\alpha x_{1}(n)+\beta x_{2}(n)\) | \(\alpha X_{1}(z)+\beta X_{2}(z)\) | At least \(\mathrm{ROC}_{1} \cap \mathrm{ROC}_{2}\) |
| Time shifing | \(x(n-k)\) | \(z^{-k}X(z)\) | \(\mathrm{ROC}\) |
| Time scaling | \(x(n/k)\) | \(X(z^k)\) | \(\mathrm{ROC}^{1/k}\) |
| Z-domain scaling | \(a^{n}x(n)\) | \(X(z/a)\) | \(|a| \: \mathrm{ROC}\) |
| Conjugation | \(\overline{x(n)}\) | \(\overline{X}(\overline{z})\) | \(\mathrm{ROC}\) |
| Convolution | \(x_{1}(n) * x_{2}(n)\) | \(X_{1}(z) X_{2}(z)\) | At least \(\mathrm{ROC}_{1} \cap \mathrm{ROC}_{2}\) |
| Differentiation in z-Domain | \([n x[n]]\) | \(-\frac{d}{d z} X(z)\) | \(\mathrm{ROC}\) = all \(\mathbb{R}\) |
| Parseval's Theorem | \(\sum_{n=-\infty}^{\infty} x[n] \mathrm{x}*[n]\) | \(\int_{-\pi}^{\pi} F(z) \mathbf{F} *(z) d z\) | \(\mathrm{ROC}\) |
