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This module will look at some of the basic properties of the ZTransform (DTFT).
The combined addition and scalar multiplication properties in the table above demonstrate the basic property oflinearity. What you should see is that if one takes the Ztransform of a linear combination of signals then itwill be the same as the linear combination of the Ztransforms of each of the individual signals. This is crucial when using a table of transforms to find the transform of a more complicated signal.
We will begin with the following signal:
Symmetry is a property that can make life quite easy when solving problems involving Ztransforms. Basicallywhat this property says is that since a rectangular function in time is a sinc function in frequency, then a sincfunction in time will be a rectangular function in frequency. This is a direct result of the symmetrybetween the forward Z and the inverse Z transform. The only difference is the scaling by $2\pi $ and a frequency reversal.
This property deals with the effect on the frequencydomain
representation of a signal if the time variable isaltered. The most important concept to understand for the
time scaling property is that signals that are narrow intime will be broad in frequency and
The table above shows this idea for the general transformation from the timedomain to the frequencydomainof a signal. You should be able to easily notice that these equations show the relationship mentioned previously: if thetime variable is increased then the frequency range will be decreased.
Time shifting shows that a shift in time is equivalent to a linear phase shift in frequency. Since the frequencycontent depends only on the shape of a signal, which is unchanged in a time shift, then only the phase spectrum willbe altered. This property is proven below:
We will begin by letting $x(n)=f(n\eta )$ . Now let's take the ztransform with the previous expression substituted in for $x(n)$ .
Convolution is one of the big reasons for converting signals to the frequency domain, since convolution in time becomesmultiplication 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 isinvolved.
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 discretetime convolution module for a more in depth explanation and derivation.
Since discrete LTI systems can be represented in terms of difference equations, it is apparent with this property that convertingto the frequency domain may allow us to convert these complicated difference equations to simpler equationsinvolving multiplication and addition.
Modulation is absolutely imperative to communications applications. Being able to shift a signal to a differentfrequency, allows us to take advantage of different parts of the electromagnetic spectrum is what allows us to transmittelevision, 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 wentthrough the steps in the previous, timeshift proof, below we will just show the initial and final step to this proof:
An interactive example demonstration of the properties is included below:
Property  Signal  ZTransform  Region of Convergence 
Linearity  $\alpha {x}_{1}\left(n\right)+\beta {x}_{2}\left(n\right)$  $\alpha {X}_{1}\left(z\right)+\beta {X}_{2}\left(z\right)$  At least ${\mathrm{ROC}}_{1}\cap {\mathrm{ROC}}_{2}$ 
Time shifing  $x(nk)$  ${z}^{k}X\left(z\right)$  $\mathrm{ROC}$ 
Time scaling  $x(n/k)$  $X\left({z}^{k}\right)$  ${\mathrm{ROC}}^{1/k}$ 
Zdomain scaling  ${a}^{n}x\left(n\right)$  $X(z/a)$  $\lefta\right\mathrm{ROC}$ 
Conjugation  $\overline{x\left(n\right)}$  $\overline{X}\left(\overline{z}\right)$  $\mathrm{ROC}$ 
Convolution  ${x}_{1}\left(n\right)*{x}_{2}\left(n\right)$  ${X}_{1}\left(z\right){X}_{2}\left(z\right)$  At least ${\mathrm{ROC}}_{1}\cap {\mathrm{ROC}}_{2}$ 
Differentiation in zDomain  $\left[nx\left[n\right]\right]$  $\frac{d}{dz}X\left(z\right)$  ROC= all $\mathbb{R}$ 
Parseval's Theorem 
$\sum_{n=()} $∞

$\int_{\pi}^{\pi} F(z)\mathrm{F*}(z)\,d z$  ROC 
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