What is the physical meaning of Z transform?

[faizalvalayam]

can anybody tell me about
what is the physical meaniing of Z transform
regards
faizalvalayam
cusat cochin

 

[islamianb4u]

z trnsform

z transform the function into a circulat form and we can detect its stability etc by just looking at the graph

 

[vadkudr]

Re: Z trnsform

Z transform allows to analyze and handle infinite discrete (in time domain) signals.
How to?
Lets do mathematical trick - multiply k-th sample on z^(-k), and sum them. We get polynomial. Infinite one in common case. Multiplication of two polynomials is convolution of their coefficients. Do you have some association? Multiplication of spectrums of signals is equivalent to their convolution. It means that by simple mathematical trick we transfered signal in some generalized frequency domain.
And indeed, Fourier transform is particular case of z transform. If z=exp(-j*w). That is values of spectrum of signal are on the unit circle of z transform. Each value is corresponding to frequency point defined by angle w=2 pi f. That's why it is called circullar frequency.

Do you familiar with Fourier transform?
You should know that Fourier transform (spectrum) of discrete (in time domain) signal is continuous and periodical.

If you see on unit circle, you can easily understand, why. Spectrum really repeats each 2pi rad if you follow circle.

So, z transform is extension of Fourier transform.
What kind of benefits we can get from this extension?
1. If we analyze signal as polynomial, we can find its zeros. It can be complex and not necessary lie on unit circle. But near them spectrum will have some deep. Closer zero, deeper notch.
2. It is interesting to analyze filters, especially IIR. IIR filter impulse response (written as infinite sequence and transformed to infinite polynomial by our trick) can be compactly represented as relation of two finite polynomials. Zeros of denominator are in fact poles of IIR transfer function in z domain. If pole is close to unit circle, it means that there huge peak in frequency response of the filter. There is property, if at least one pole is outside of unit circle - IIR filter is unstable.
3. We can make fast convolution algorithms for finite signal. We can choose several arbitrary points on z-plain (complex, or real - your choice) and calculate values of our signal-polynomials. After pointwise multiplying of these generalized spectrum values, we can recover polynomial that is convolution of our signal. If you choose n points, you can recover polynomial (that is signal) up to power n-1 (n samples of signal). Indeed Fourier is this kind of algorithm with points z=exp(-j*2*pi/n) equidistantly placed on unit circle. But complex number is not convenient. Really, there exist fast algorithms with no complex calculations.

There are only small essay :-( It is impossible to cover all cases in one message. Anyway, z transform is powerful tool for handling discrete linear signals and systems

 

[hamran]

Re: Z trnsform

the z transform is a helpful tool in the analysis the copmlex discrete signals as it makes that much easier.

 

[naresh850]

Z trnsform

As laplac transform is useful for analycis for analog signal, The Z transform is useful for analysis of discreate transform.i.e stability of system