This module will look at some of the basic properties of the ZTransform (DTFT).
Inside Collection (Course): Signals and Systems
Summary: This module includes a table of important properties of the ztransform.
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 of linearity. What you should see is that if one takes the Ztransform of a linear combination of signals then it will 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. 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
This property deals with the effect on the frequencydomain 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 infinitelength constant function in frequency.
The table above shows this idea for the general transformation from the timedomain to the frequencydomain 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 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:
We will begin by letting
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 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 converting to the frequency domain may allow us to convert these complicated difference equations to simpler equations involving multiplication and addition.
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, 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:
Download LabVIEW Source 
Property  Signal  ZTransform  Region of Convergence 
Linearity 


At least 
Time shifing 



Time scaling 



Zdomain scaling 



Conjugation 



Convolution 


At least 
Differentiation in zDomain 


ROC= all 
Parseval's Theorem 


ROC 
"My introduction to signal processing course at Rice University."