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What is the relationship between z-transform and frequency response of the signal?

Started by runinrainy 7 months ago4 replieslatest reply 7 months ago264 views

Hello,

I am trying to understand  intuitively the relationship of the zeros of the z-transform of a discrete time signal and it's frequency domain representation. Let's say I have a finite duration of time signal, and by taking it's Z-Transform and Fourier Transform, I want to see the relation between zeros in the z-domain and frequency components in the frequency domain. How can we interpret the zeros in z- domain  and the notches or fades in the frequency domain?

 Many thanks in advance!





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Reply by ing_jpuDecember 26, 2023

First, the Fourier Transform (the frequency response of the system) can be recovered from the z-domain by evaluating the z-transform at the unit circumference with its center at the origin. In other words, all the complex numbers with unit modulus within the z-plane correspond to the Fourier Transform. Secondly, remember that linear and time-invariant systems have a rational function as transfer function, namely the ratio of a couple of polynomials. Since any polynomial can be factorized according to its roots, such a representation will lead to factors of the form (z - r_k), where r_k is k-th root (zero or pole, depending on wheter you are dealing with the numerator or denominator polynomial). You can interpret this factor (z - r_k) as a vector starting at r_k and ending at z. As the frequency response of the system is obtained by adopting z=exp(jw), the factor (z - r_k) represents the vector whose tail is located at the root r_k (zeroes or poles) and its head is located at the unit circumference. The modulus of the transfer function results in |z - r_k|, representing the distance from the roots to the unit circumference. Therefore, when zeros are located at the unit circumference, the distance is 0, creating a notch at that frequency w. If the zeros are not located on the unit circumference, they result in non-zero valleys at the given frequency w.

For example, a FIR system: H(z)= 1-bz^{-1} = (z - b)/z, has a zero at z=b. If b = exp(j pi/3) (which is located on the unit circumference), it results in a notch at the angular frequency w=pi/3. In addition, as the pole is z=0, you will observe a "peak" at the angular frequency w=0. Therefore, this FIR system behaves as a low-pass. However, this system notches only at the positive frequency w=+pi/3, and the output of the system will be complex when the input is real. Nevertheless, its simplicity can be utilized to provide some intuition for your inquiry.

Best regards

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Reply by runinrainyDecember 27, 2023

Many thanks for the detailed answer! Really helps. 

Lets imagine this as a wireless communication system. Basically, transmitting the time signal over the wireless channel which is like an FIR filter. So the z-transform of the received signal has both channel zeros and transmitted signal zeros. 


1- Suppose that there is an RF impairments at the transmitter where the carrier frequency offset  is introduced, in other words, there is a frequency shift already experienced by the transmitted signal. So in this case, how can we interpret the zeros of the z-transform of the received signal? How are the channel zeros and transmitted signal zeros affected by this frequency offset?


2- Assume there is a mobility or doppler effect in the channel, along with the RF impairments at both Transmitter and Receiver. What is the effect of these frequency offsets on the zeros of the z-transform of the transmitted signal?


Best Regards,

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Reply by kazDecember 28, 2023

This link may help. (noting that I always see people using DFT for FIR and DFT or Z for IIR except University lectures that stress on Z)

https://uk.mathworks.com/videos/understanding-the-...


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Reply by Tim WescottDecember 28, 2023

... where the carrier frequency offset  is introduced ...

Carrier frequency offsets can only be introduced by time-varying systems, which translate to shift-varying systems when you sample the data.  The z transform is really only useful for shift-invariant systems.

So the z transform is not useful in this circumstance.

You can use the discrete-time Fourier transform for this, but you need to apply it with care.

You have some underlying question here -- it may be best to start a new thread, describe the problem that's actually giving you pain, and ask how to solve it without restricting those solutions to a particular solution space.