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!

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

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,

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)

... 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.