22. Z-transform

Contents

• Definition

• Elementary transforms

• Region of convergence

• Applications

  • Solution of difference equations

  • Analysis of stability of numerical integration methods

  • Discrete-time design of filters and discrete-time systems. Relation between continuous-time and discrete-time systems due to sampling. Relation with Laplace transform.

22.1. Definition

Z-transform can be considered as a discrete-time counterpart of the Laplace transform, as section Laplace transform of a sampled function tries to show. Z-transform converts a discrete-time signal \(\{ x_n \}\) into a complex-valued function \(X(z)\). Similarly to the definition of Laplace transform, Z-transform can be either two-sided

\[X(z) = \mathscr{Z}\{ x_n \}(z) = \sum_{n=-\infty}^{+\infty} x_n z^{-n} \ ,\]

or one-sided

\[X(z) = \mathscr{Z}\{ x_n \}(z) = \sum_{n=0}^{+\infty} x_n z^{-n} \ ,\]

that can be also thought as the two-sided transform of causal time-series, i.e. \(x_{k} = 0\), \(\forall k < 0\).

22.1.1. Z-transform and discrete-time Fourier transform (DTFT)

The discrete-time Fourier transform of a sequence, that can be written as (21.6),

\[\text{DTFT}(x(t); \Delta t)(\nu) = \Delta t \sum_{n=-\infty}^{+\infty} x(n \Delta t) e^{-i 2 \pi \nu n \Delta t} \ ,\]

can be thought as a scaled Z-transform of the signal, evaluated in \(z = i \omega \Delta t = i 2 \pi \nu \Delta t\),

\[\text{DTFT}(x(t); \Delta t)(\nu) = \Delta t \left.\left[ \sum_{n=-\infty}^{+\infty} x_n z^{-n} \right]\right|_{z=e^{i \omega \Delta t}} = \Delta t X(z=e^{i \omega \Delta t}) \ ,\]

22.1.2. Laplace transform of a sampled function

Delta sampling. Let \(\widetilde{x}(t) = x(t) \text{III}_{\delta t}(t) = x(t) \sum_{m = - \infty}^{+\infty} \delta(t - m \Delta t)\) the sampling of the continuous-time function \(x(t)\) with Dirac’s comb. Its one-sided Laplace transform reads

(22.1)\[\begin{split}\begin{aligned} \mathscr{L}\{ \widetilde{x}(t) \}(s) & = \int_{t=0^-}^{+\infty} \widetilde{x}(t) e^{-s t} dt = \\ & = \sum_{m=-\infty}^{+\infty} \int_{t=0^-}^{+\infty} x(t) \delta( t - m \Delta t ) e^{-s t} dt = \\ & = \sum_{m= 0 }^{+\infty} x(m \Delta t) e^{-s m \Delta t} = \\ & = \sum_{m= 0 }^{+\infty} x(t_m) e^{-s t_m} = \\ & = \sum_{m= 0 }^{+\infty} x_m z^{-m} \ , \end{aligned}\end{split}\]

with \(x_m = x(t_m)\), \(t_m = m \Delta t\), and \(z = e^{s \Delta t}\). Here the summation runs over \(m = 0:+\infty\), as the integral of \(\delta(t - m \Delta t)\) produces a non-zero result only for \(m \ge 0\). The relation (22.1) shows that the Laplace transform of the Dirac delta sampled function is equivalent to the Z-transform of its samples, if the relations between \(t_m\) and \(m\) and \(z\) and \(s \Delta t\) hold.

Sample and Hold.

(22.2)\[\begin{split}\begin{aligned} \mathscr{L}\{ \widetilde{x}(t) \}(s) & = \int_{t=0^-}^{+\infty} \widetilde{x}(t) e^{-s t} dt = \\ & = \sum_{m=-\infty}^{+\infty} \int_{t=t_m}^{t_{m+1}} x(t_m) e^{-s t} dt = \\ & = \sum_{m=-\infty}^{+\infty} x(t_m) \frac{1}{s} \left[ e^{-s t_m} - e^{-s t_{m+1}} \right] = \\ & = \frac{1 - e^{-s \Delta t}}{s} \sum_{m=-\infty}^{+\infty} x_m e^{-s t_m} = \\ & = \dots \end{aligned}\end{split}\]

with \(z = e^{s \Delta t}\), \(s = \frac{1}{\Delta t} \ln z\).

22.2. Inverse transform

…

22.3. Elementary one-sided transforms

In this section, some elementary Z-transforms are collected. These can be interpreted as the discrete-time counterpart of the elementary Laplace transforms

Linearity.

Kronecker delta. \(x_n = \delta_{0n} = \left\{ \begin{aligned} & 1 && n = 0 \\ & 0 && n \ne 0 \end{aligned}\right.\)

\[\mathscr{Z}\{ \delta_{0n} \}(z) = \sum_{n=0}^{+\infty} \delta_{n0} z^{-n} = 1 \ .\]

“Time” delay. Z-transform of a causal sequence \(x_n\) (with \(x_n = 0\) for \(n < 0\)), \(x_{n-k}\) with \(k \ge 0\)

\[ \mathscr{Z}\{ x_{n-k} \}(z) = z^{-k} X(z) \]

Proof

\[\begin{split}\begin{aligned} \mathscr{Z}\{ x_{n-k} \}(z) & = \sum_{n=0}^{+\infty} x_{n-k} z^{-n} = \\ & = \sum_{n=k}^{+\infty} x_{n-k} z^{-(n-k)} z^{-k} = \\ & = \sum_{n=0}^{+\infty} x_{n} z^{-n} z^{-k} = \\ & = z^{-k} X(z) \ . \end{aligned}\end{split}\]

First difference backward. Directly follows from the Z-transform of time delay

\[\mathscr{Z}\left\{ x_n - x_{n-1} \right\} = \left( 1 - z^{-1} \right) X(z)\]

First difference forward.

Accumulation.

Convolution.

Initial value. If \(x_n\) is causal, i.e. \(x_n = 0\) for \(n < 0\), then

\[x_{0} = \lim_{z \rightarrow +\infty} X(z) \ .\]

Proof

\[X(z) = x_0 + \sum_{n=1}^{+\infty} x_n z^{-n}\]

As

\[\left|\sum_{n=1}^{+\infty} x_n z^{-n} \right| \le \sum_{n=1}^{+\infty} \left| x_n z^{-n} \right| \le M \sum_{n=1}^{+\infty} |z|^{-n} = M \frac{1}{|z|} \frac{1}{1 - \frac{1}{|z|}} \ ,\]

for every \(z\), with \(|z| > 1\) s.t. the geometric series is convergent. As \(|z| \rightarrow + \infty\), the series goes to zero, and thus

\[\lim_{z \rightarrow +\infty} X(z) = x_0 \ .\]

Final value. If the poles of \((z-1) X(z)\) are inside the unit circle (condition for as.stability), then

\[x_{+\infty} = \lim_{z \rightarrow 1} (z-1) X(z) \ .\]

Proof

…

22.4. Applications

22.4.1. Solution of difference equations

First-order reduction.

Reduction to first order \(\ k\)-dimensional system of a \(\ k\)-order ODE

From \(k\)-order difference equation to system of \(1\)-order difference equations, as it can be done with ODEs. For ODEs

\[x^{(k)} + a_{k-1} x^{(k-1)} + \dots + a_1 \dot{x} + a_0 x = f(t)\]

may be written as a first-order \(k\)-dimensional system

\[\dot{\mathbf{z}} = \mathbf{f}(\mathbf{z},t) \ \]

\[\begin{split}\begin{bmatrix} \dot{z}_0 \\ \dot{z}_1 \\ \dots \\ \dot{z}_{k-1} \end{bmatrix} = \begin{bmatrix} z_1 \\ z_2 \\ \dots \\ - a_{k-1} z_{k-1} - \dots - a_1 z_1 - a_0 z_0 + f(t) \end{bmatrix}\end{split}\]

Reduction to first order \(\ k\)-dimensional system of a \(\ k\)-order difference equation

For a difference equation

\[x_{n} + a_{1} x_{n-1} + \dots + a_{k-1} x_{n-k+1} = f_n\]

may be written as

\[\mathbf{z}_{n+1} = \mathbf{f}(\mathbf{z}_n, t_n)\]

\[\begin{split}\begin{bmatrix} z_{0,n} \\ z_{1,n} \\ \dots \\ z_{k-1,n} \end{bmatrix} = \begin{bmatrix} z_{1,n-1} \\ z_{2,n-1} \\ \dots \\ f_n - a_{1} z_{k-1,n-1} \dots - a_{k-1} z_{0,n-1} \end{bmatrix}\end{split}\]

with \(z_{j,n} = x_{n-k+1+j}\), for \(j=0:k-1\).

Ansatz. The solution of a homogeneous linear ODE can be solved as a linear combination of elementary solutions with expression \(x(t) = X e^{st}\) - except for the case of multiple roots in the characteristic polynomial

\[x^{(k)} + a_{k-1} x^{(k-1)} + \dots + a_1 \dot{x} + a_0 x = 0 \ , \]

with values of \(s\) as the roots of the polynomial

\[s^k + a_{k-1} s^{k-1} + \dots + a_1 s + a_0 = 0 \ .\]

The solution is then

\[x(t) = \sum_{a = 1}^{k} X_a e^{s_a t} \ ,\]

with the integration constants to be determined with the initial conditions provided along with the differential equation. Asymptotical stability conditions depend on the real part of the eigenvalues, and on the multiplicity of the imaginary eigenvalues.

A linear difference equation can be solved as a linear combination of elementary solutions with expression \(x_n = X z^n\)

\[x_{n} + a_{k-1} x_{n-1} + \dots + a_1 x_{n-k+2} + a_0 x_{n-k+1} = 0 \ , \]

with values of \(z\) as the roots of the polynomial

\[z^k + a_{k-1} z^{k-1} + \dots + a_1 z + a_0 = 0 \ .\]

The solution is then

\[x_n = \sum_{a = 1}^{k} X_a {z_a}^{n} \ ,\]

with the integration constants to be determined with the initial conditions provided along with the difference equation.

Stability conditions. …

• asymptotically stable if all the eigenvalues are \(|z_a| < 1\)

• unstable if…

• marginally stable if…

todo treat multiple roots

22.4.2. Analysis of stability of numerical integration methods for ODEs