Z Transform

Z-Transform Analysis of Time Series Models

The z-transform is a mathematical tool that converts a discrete-time signal (like a time series) into a complex frequency-domain representation. It is the discrete-time equivalent of the Laplace transform and is instrumental in analyzing the properties of time series models.

Z-Transform Representation

Using the backshift operator B, where BX_t = X_t-1, the ARMA(p,q) model can be written in polynomial form:

(1 - φ₁B - ... - φ_pB^p)X_t = c + (1 + θ₁B + ... + θ_qB^q)ε_t

Let Φ(B) and Θ(B) be the polynomials in the backshift operator. Replacing B with z^-1 gives the z-transform representation:

Φ(z^-1)X(z) = c' + Θ(z^-1)E(z)

where X(z) and E(z) are the z-transforms of the time series and the error term, respectively.

Transfer Function

The transfer function, H(z), of an ARMA model describes the relationship between the input (error term) and the output (time series) in the z-domain. It is defined as the ratio of the MA polynomial to the AR polynomial:

H(z) = X(z) / E(z) = Θ(z^-1) / Φ(z^-1)

• For a pure AR(p) model, the transfer function is H(z) = 1 / Φ(z^-1), which is an all-pole function.
• For a pure MA(q) model, the transfer function is H(z) = Θ(z^-1), which is an all-zero function.
• An ARMA(p,q) model has a pole-zero transfer function.

Stability and Invertibility Conditions

The stability of an ARMA model is determined by the roots of the autoregressive polynomial, Φ(z). For a model to be stable (and thus stationary), all the roots of Φ(z) must lie outside the unit circle in the z-plane. This is equivalent to the poles of the transfer function H(z) lying inside the unit circle when expressed in terms of z.

The invertibility of an ARMA model is determined by the roots of the moving average polynomial, Θ(z). For the model to be invertible, all the roots of Θ(z) must lie outside the unit circle. Invertibility ensures that the model can be represented as a pure autoregressive process of infinite order.

 The discrete-time (DT) signal, which is a series of real or complex numbers, is transformed into a complex frequency-domain (z-domain or z-plane) representation using the Z-transform in signal processing.

Z Transform of a delta or unit impulse function

Example of Z Transform

For a real world example, when we send a unit impulse signal for testing input we receive multiple impulse responses at receiver due to different multipath. 

Let's assume, data signal x[n] = [2   -5    1    3]

and channel impulse responses h[n] = [-1  4   2]

Now simply multiply the data signal and channel co-efficients learned in elementary school

3  1  -5   2

    2   4  -1

-----------------------

6  14  -9   -17   13    -2

It can be represented as

x[n]*h[n] = 6z^(-5) + 14z^(-4)  - 9z^(-3)  - 17z^(-2)  +  13z^(-1)  -  2z

After computing discrete time z transform it is defined as simple multiplication of X(z) and H(z).