Signals and Systems

10.5 Properties of the Z-Transform

Imron Rosyadi

10.5 Properties of the Z-Transform

Just like with the Fourier and Laplace transforms, the Z-transform comes with a powerful set of properties. These properties simplify the analysis and design of discrete-time signals and systems, allowing us to manipulate transforms algebraically rather than dealing with complex convolutions or direct Z-transform definitions. Understanding these properties is crucial for efficient problem-solving in ECE.

10.5.1 Linearity

Property:

If \(x_1[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X_1(z)\) with ROC \(R_1\), and \(x_2[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X_2(z)\) with ROC \(R_2\), then \[ a x_1[n] + b x_2[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} a X_1(z) + b X_2(z) \] ROC: Contains \(R_1 \cap R_2\).

Implications for ROC:

• If no pole-zero cancellation occurs, the ROC is exactly \(R_1 \cap R_2\).
• If pole-zero cancellation occurs, the ROC can be larger than \(R_1 \cap R_2\).
  • Example: \(a^n u[n]\) has ROC \(|z|>|a|\).
  • \(-a^n u[-n-1]\) has ROC \(|z|<|a|\).
  • But their sum, \(\delta[n]\), has ROC as the entire \(z\)-plane.

Application Example:

Suppose you have a system that processes two signals independently and then sums their outputs.

If \(y_1[n] = H_1(z)X_1(z)\) and \(y_2[n] = H_2(z)X_2(z)\),

then \(y[n] = y_1[n] + y_2[n]\) has \(Y(z) = Y_1(z) + Y_2(z)\).

Linearity is perhaps the most fundamental property. It means that the Z-transform of a sum of scaled signals is the sum of their scaled Z-transforms. This is incredibly useful because it allows us to break down complex signals into simpler components, transform each component, and then recombine them in the Z-domain.

The ROC aspect is important: the ROC of the sum is at least the intersection of the individual ROCs. However, it can be larger if pole-zero cancellations occur, as seen in the example of summing two infinite-duration sequences to get a finite-duration one (like \(\delta[n]\)).

10.5.2 Time Shifting

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\), then \[ x[n-n_0] \stackrel{\mathcal{Z}}{\longleftrightarrow} z^{-n_0} X(z) \] ROC: \(R\), except for possible addition/deletion of \(z=0\) or \(z=\infty\).

Explanation of ROC change:

• If \(n_0 > 0\) (delay): \(z^{-n_0}\) introduces poles at \(z=0\). If \(X(z)\) had a zero at \(z=0\) that cancels these poles, the ROC might gain \(z=0\). If \(X(z)\) did not have a zero, \(z=0\) might be excluded from the ROC of \(z^{-n_0}X(z)\).
• If \(n_0 < 0\) (advance): \(z^{-n_0}\) introduces zeros at \(z=0\). This is equivalent to multiplying by \(z^{|n_0|}\), which introduces poles at \(z=\infty\). Similarly, \(z=\infty\) might be excluded from the ROC.

10.5.2 Time Shifting

Application Example:

Consider a simple discrete-time filter defined by the difference equation:

\(y[n] = x[n] - x[n-1]\)

Applying the Z-transform to each term:

\(Y(z) = X(z) - z^{-1}X(z)\)

\(Y(z) = (1 - z^{-1})X(z)\)

Here, \(H(z) = 1 - z^{-1}\), which is the Z-transform of \(h[n]=\delta[n]-\delta[n-1]\).

This property is fundamental for converting difference equations to algebraic equations in the Z-domain.

Time shifting is another very intuitive property. Delaying a signal by \(n_0\) samples in the time domain corresponds to multiplying its Z-transform by \(z^{-n_0}\). This algebraic operation in the Z-domain is much simpler than handling delays in the time domain, especially for complex systems.

The changes in ROC at \(z=0\) or \(z=\infty\) are subtle but important to remember, as they relate to whether the original signal had poles/zeros at those locations that might be canceled or introduced by the \(z^{-n_0}\) term. The interactive plot shows a simple signal and its delayed version.

10.5.3 Scaling in the \(z\)-Domain

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\), then \[ z_0^n x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X\left(\frac{z}{z_0}\right) \] ROC: \(|z_0|R\). If \(z\) is in \(R\), then \(|z_0|z\) is in the new ROC.

Special Case: Frequency Shifting (\(z_0=e^{j\omega_0}\))

\[ e^{j\omega_0 n} x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(e^{-j\omega_0}z) \]

• ROC remains \(R\) (since \(|e^{j\omega_0}|=1\)).
• Poles and zeros of \(X(z)\) at \(z=a\) are rotated to \(z=a e^{j\omega_0}\) in \(X(e^{-j\omega_0}z)\).
• This corresponds to a frequency shift in the DTFT.

Application Example: Demodulation

In digital communication systems, a signal might be modulated by multiplying it with a complex exponential. This property allows us to analyze the effect of this modulation (or demodulation) directly in the Z-domain.

Scaling in the \(z\)-domain, which means multiplying \(x[n]\) by an exponential \(z_0^n\), has a profound effect on the Z-transform and its pole-zero plot. It scales the ROC and rotates/scales the pole-zero locations.

The most common application is frequency shifting (when \(z_0\) is a complex exponential \(e^{j\omega_0}\)). This rotates the entire pole-zero pattern in the Z-plane. This property is crucial in communication systems for tasks like modulation and demodulation, where signals are shifted in frequency.

10.5.4 Time Reversal

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\), then \[ x[-n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X\left(\frac{1}{z}\right) \] ROC: \(1/R\). If \(z_0\) is in \(R\), then \(1/z_0\) is in the ROC for \(x[-n]\).

Implications:

• If \(X(z)\) has a pole at \(z=a\), then \(X(1/z)\) has a pole at \(z=1/a\).
• This means the pole-zero pattern is inverted with respect to the unit circle.
  • Poles/zeros inside the unit circle move outside.
  • Poles/zeros outside the unit circle move inside.
  • Poles/zeros on the unit circle stay on the unit circle.

Application Example: Useful in analyzing systems that process signals in reverse, or for understanding properties of symmetric signals.

Consider \(x[n] = a^n u[n]\): \(X(z) = \frac{1}{1-az^{-1}}\), ROC \(|z|>|a|\).

Then \(x[-n] = a^{-n} u[-n]\) (note: \(u[-n]\) is a left-sided sequence starting at \(n=0\)).

\(x[-n]\) is related to \(a^{-n} u[-n-1]\) which has \(Z\)-transform \(\frac{-1}{1-a^{-1}z^{-1}}\), ROC \(|z|<|a|^{-1}\).

The property \(X(1/z)\) applied to \(X(z)\) gives \(\frac{1}{1-a(1/z)^{-1}} = \frac{1}{1-az}\).

This is the \(Z\)-transform of \(x[-n]\) only if the ROC is specified correctly.

Time reversal flips the signal in time. In the Z-domain, this corresponds to replacing \(z\) with \(1/z\). This operation has a direct impact on the pole-zero locations: each pole or zero at \(a\) moves to \(1/a\). This means poles inside the unit circle move outside, and vice versa. The ROC also gets inverted. This property helps analyze signals that are symmetric or when dealing with concepts like minimum-phase systems.

10.5.5 Time Expansion

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\),

and \(x_{(k)}[n]\) is defined as \(x[n/k]\) if \(n\) is a multiple of \(k\), and \(0\) otherwise (inserting \(k-1\) zeros). Then \[ x_{(k)}[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z^k) \] ROC: \(R^{1/k}\). If \(z_0\) is in \(R\), then \(z_0^{1/k}\) is in the new ROC.

Explanation:

From \(X(z) = \sum x[n]z^{-n}\),

\(X(z^k) = \sum x[n](z^k)^{-n} = \sum x[n]z^{-kn}\).

This sum only contains terms \(z^{-m}\) where \(m\) is a multiple of \(k\). The coefficient of \(z^{-kn}\) is \(x[n]\), which is \(x_{(k)}[kn]\). The coefficients of \(z^{-m}\) where \(m\) is not a multiple of \(k\) are zero.

10.5.5 Time Expansion

Application Example:

• Interpolation/Decimation: This property is crucial in multirate signal processing, where signals are upsampled (interpolation by inserting zeros) or downsampled.
• Filter Banks: Design of filter banks often involves time expansion and compression operations.

Consider \(x[n]=\delta[n] + \delta[n-1]\). \(X(z)=1+z^{-1}\).

If \(k=2\), \(x_{(2)}[n]=\delta[n] + \delta[n-2]\).

Then \(X(z^2) = 1+(z^2)^{-1} = 1+z^{-2}\). This matches.

Time expansion, or upsampling, involves inserting zeros between samples of a discrete-time signal. In the Z-domain, this corresponds to replacing \(z\) with \(z^k\). This property is fundamental in multirate signal processing, which is used extensively in audio, image, and communication systems to change sampling rates. The interactive plot shows how a signal looks after inserting zeros.

10.5.6 Conjugation

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\),

then \[ x^*[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X^*(z^*) \] ROC: \(R\).

Implications for Real Signals:

If \(x[n]\) is real (\(x[n]=x^*[n]\)), then \(X(z) = X^*(z^*)\).

This implies that if \(X(z)\) has a pole (or zero) at \(z=z_0\), it must also have a pole (or zero) at its complex conjugate \(z=z_0^*\).

• Complex poles/zeros always appear in conjugate pairs for real signals.
• This is why we always see complex conjugate poles in second-order systems with real coefficients.

Application Example:

When designing filters with real coefficients (which is almost always the case in practice), this property guarantees that if you place a complex pole or zero, its conjugate must also be present. This simplifies design by reducing the number of independent parameters.

Conjugation is a straightforward property. If you take the complex conjugate of a signal, its Z-transform becomes the complex conjugate of \(X(z^*)\).

The most important consequence is for real-valued signals. Since most physical signals are real, this property tells us that all complex poles and zeros in the Z-plane must appear in conjugate pairs. This simplifies filter design and analysis significantly because you only need to specify one of the complex conjugate pair.

10.5.7 The Convolution Property

Property:

If \(x_1[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X_1(z)\) with ROC \(R_1\), and \(x_2[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X_2(z)\) with ROC \(R_2\), then \[ x_1[n] * x_2[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X_1(z) X_2(z) \] ROC: Contains \(R_1 \cap R_2\).

Significance:

• Convolution in the time domain becomes simple multiplication in the Z-domain.
• This is the cornerstone for analyzing LTI systems using Z-transforms.
  • Output \(Y(z) = H(z)X(z)\).

Implications for ROC:

• As with linearity, if pole-zero cancellation occurs in \(X_1(z)X_2(z)\), the ROC can be larger than \(R_1 \cap R_2\).

10.5.7 The Convolution Property

Application Example: LTI Systems

For an LTI system with impulse response \(h[n]\) and input \(x[n]\), the output is \(y[n] = h[n] * x[n]\).

In the Z-domain, this becomes \(Y(z) = H(z)X(z)\).

Example 10.15: First Difference

\(y[n] = x[n] - x[n-1]\)

\(Y(z) = X(z) - z^{-1}X(z) = (1-z^{-1})X(z)\)

So, \(H(z) = 1-z^{-1}\). This is the Z-transform of \(h[n]=\delta[n]-\delta[n-1]\).

Example 10.16: Accumulation (Running Sum)

\(w[n] = \sum_{k=-\infty}^{n} x[k] = u[n] * x[n]\)

\(W(z) = U(z)X(z) = \frac{1}{1-z^{-1}} X(z)\)

So, \(H(z) = \frac{1}{1-z^{-1}}\). This is the Z-transform of \(h[n]=u[n]\).

The convolution property is arguably the most important property of the Z-transform for system analysis. It transforms complex time-domain convolutions into simple algebraic multiplications in the Z-domain. This greatly simplifies the analysis of LTI systems, allowing us to find system responses by merely multiplying transfer functions and input transforms.

It also provides insights into operations like differencing and accumulation, showing their corresponding Z-domain representations.

10.5.8 Differentiation in the \(z\)-Domain

Property:

If \(x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)\) with ROC \(R\), then \[ n x[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} -z \frac{dX(z)}{dz} \] ROC: \(R\).

Derivation:

Start from \(X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}\).

Differentiate with respect to \(z\):

\(\frac{dX(z)}{dz} = \sum_{n=-\infty}^{+\infty} x[n] (-n) z^{-n-1} = -z^{-1} \sum_{n=-\infty}^{+\infty} n x[n] z^{-n}\) Multiply by \(-z\):

\(-z \frac{dX(z)}{dz} = \sum_{n=-\infty}^{+\infty} n x[n] z^{-n} = \mathcal{Z}\{n x[n]\}\)

10.5.8 Differentiation in the \(z\)-Domain

Application Example 10.17:

Find \(x[n]\) for \(X(z)=\log(1+az^{-1}), |z|>|a|\).

We know \(\mathcal{Z}\{nx[n]\} = -z \frac{dX(z)}{dz}\).

\(\frac{dX(z)}{dz} = \frac{1}{1+az^{-1}} (-a z^{-2}) = \frac{-a z^{-2}}{1+az^{-1}}\) So, \(\mathcal{Z}\{nx[n]\} = -z \left( \frac{-a z^{-2}}{1+az^{-1}} \right) = \frac{a z^{-1}}{1+az^{-1}}\).

Now, we need to find the inverse Z-transform of \(\frac{a z^{-1}}{1+az^{-1}}\).

We know \(\frac{1}{1+az^{-1}} \stackrel{\mathcal{Z}}{\longleftrightarrow} (-a)^n u[n]\) (for \(|z|>|a|\)).

Using time shifting (\(z^{-1}\)): \(\frac{az^{-1}}{1+az^{-1}} \stackrel{\mathcal{Z}}{\longleftrightarrow} a(-a)^{n-1} u[n-1]\). So, \(nx[n] = a(-a)^{n-1}u[n-1]\).

\(x[n] = \frac{a(-a)^{n-1}}{n} u[n-1] = \frac{(-a)(-a)^{n-1}}{n} u[n-1] = \frac{-(-a)^n}{n} u[n-1]\).

This matches Example 10.14.

Differentiation in the Z-domain relates to multiplication by \(n\) in the time domain. This property is very useful for finding the Z-transforms of signals like \(n a^n u[n]\) or for solving certain types of difference equations.

The example demonstrates how we can use this property to inverse Z-transform a seemingly complex function like a logarithm by first differentiating it, which often turns it into a rational function that can be handled by other methods like partial-fraction expansion.

10.5.9 The Initial-Value Theorem

Property:

If \(x[n]=0\) for \(n<0\) (i.e., \(x[n]\) is causal), then \[ x[0] = \lim_{z \rightarrow \infty} X(z) \] Derivation:

For a causal sequence, \(X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} = x[0]z^0 + x[1]z^{-1} + x[2]z^{-2} + \dots\)

As \(z \rightarrow \infty\):

• \(z^{-n} \rightarrow 0\) for \(n>0\).
• \(z^0 = 1\).

So, \(\lim_{z \rightarrow \infty} X(z) = x[0]\).

10.5.9 The Initial-Value Theorem

Application Example 10.19:

Consider \(X(z)=\frac{1-\frac{3}{2} z^{-1}}{\left(1-\frac{1}{3} z^{-1}\right)\left(1-\frac{1}{2} z^{-1}\right)}\) from Example 10.3 (where \(x[n]\) was causal).

\(\lim_{z \rightarrow \infty} X(z) = \lim_{z \rightarrow \infty} \frac{1-\frac{3}{2} z^{-1}}{1-\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\)

As \(z \rightarrow \infty\), all terms with \(z^{-1}\) or \(z^{-2}\) go to zero.

So, \(\lim_{z \rightarrow \infty} X(z) = \frac{1-0}{1-0+0} = 1\).

This is consistent with \(x[0]=1\) found in Example 10.3.

Consequence:

For a causal \(X(z)\), if \(x[0]\) is finite, then \(\lim_{z \rightarrow \infty} X(z)\) must be finite. This means the number of finite zeros of \(X(z)\) cannot be greater than the number of finite poles.

The initial-value theorem provides a quick way to find the first sample (\(x[0]\)) of a causal signal directly from its Z-transform. It’s a useful check for calculations and provides insight into the behavior of the Z-transform at infinity. For causal systems, if \(x[0]\) is finite, it also implies a constraint on the number of poles and zeros.

Summary of Z-Transform Properties

Property	Time Domain \(x[n]\)	Z-Domain \(X(z)\)	ROC
Linearity	\(ax_1[n]+bx_2[n]\)	\(aX_1(z)+bX_2(z)\)	Contains \(R_1 \cap R_2\)
Time Shifting	\(x[n-n_0]\)	\(z^{-n_0}X(z)\)	\(R\) (possibly \(\pm 0, \pm \infty\))
Scaling in \(z\)-Domain	\(z_0^n x[n]\)	\(X(z/z_0)\)	\(|z_0|R\)
Time Reversal	\(x[-n]\)	\(X(1/z)\)	\(1/R\)
Time Expansion	\(x_{(k)}[n]\)	\(X(z^k)\)	\(R^{1/k}\)
Conjugation	\(x^*[n]\)	\(X^*(z^*)\)	\(R\)
Convolution	\(x_1[n]*x_2[n]\)	\(X_1(z)X_2(z)\)	Contains \(R_1 \cap R_2\)
Differentiation in \(z\)-Domain	\(nx[n]\)	\(-z \frac{dX(z)}{dz}\)	\(R\)
Initial-Value Theorem	\(x[0]\) (for causal \(x[n]\))	\(\lim_{z \rightarrow \infty} X(z)\)	N/A (requires causal \(x[n]\))

Summary of Z-Transform Properties

Why these properties are important:

• Simplification: They convert complex time-domain operations (convolution, differentiation, shifting) into simpler algebraic manipulations in the Z-domain.
• System Analysis: Essential for analyzing the behavior of LTI systems (e.g., finding impulse responses, transfer functions, stability).
• Filter Design: Used to design digital filters by relating desired time-domain characteristics (e.g., causality) or frequency-domain characteristics (e.g., frequency shifting) to Z-domain manipulations.
• Problem Solving: Provide powerful tools for solving difference equations and finding inverse Z-transforms.

Important

The Z-transform properties are the core tools that make the Z-transform a powerful and indispensable analytical method in discrete-time signal processing and control systems.

This summary table is a quick reference for all the properties we’ve covered. It’s important to not just memorize them, but to understand what each property means intuitively and how it relates to real-world signals and systems.

These properties are the reason the Z-transform is so valuable in ECE. They provide a mathematical framework that simplifies complex problems, allowing us to analyze, design, and implement discrete-time systems much more efficiently.

10.6 SOME COMMON z-TRANSFORM PAIRS

TABLE 10.1 PROPERTIES OF THE \(z\)-TRANSFORM

10.6 SOME COMMON z-TRANSFORM PAIRS

TABLE 10.2 SOME COMMON \(z\)-TRANSFORM PAIRS

Signal	Transform	ROC
1. \(\delta[n]\)	1	All \(z\)
2. \(u[n]\)	\(\frac{1}{1-z^{-1}}\)	\(\|z\|>1\)
3. \(-u[-n-1]\)	\(\frac{1}{1-z^{-1}}\)	\(\|z\|<1\)
4. \(\delta[n-m]\)	\(z^{-m}\)	All \(z\), except
0 if \(m>0)\) or
		\(x\) (if \(m<0)\)
5. \(\alpha^{n} u[n]\)	\(\frac{1}{1-\alpha z^{-1}}\)	\(\|z\|>\|\alpha\|\)
6. \(-\alpha^{n} u[-n-1]\)	\(\frac{1}{1-\alpha z^{-1}}\)	\(\|z\|<\|\alpha\|\)
7. \(n \alpha^{n} u[n]\)	\(\frac{\alpha z^{-1}}{\left(1-\alpha z^{-1}\right)^{2}}\)	\(\|z\|>\|\alpha\|\)
8. \(-n \alpha^{n} u[-n-1]\)	\(\frac{\alpha z^{-1}}{\left(1-\alpha z^{-1}\right)^{2}}\)	\(\|z\|<\|\alpha\|\)
9. \(\left[\cos \omega_{0} n\right] u[n]\)	\(\frac{1-\left[\cos \omega_{0}\right] z^{-1}}{1-\left[2 \cos \omega_{0}\right] z^{-1}+z^{-2}}\)	\(\mid z 1\)
10. \(\left[\sin \omega_{0} n\right] u[n]\)	\(\frac{\left[\sin \omega_{0}\right] z^{-1}}{1-\left[2 \cos \omega_{0}\right] z^{-1}+z^{-2}}\)	\(\|z\|>1\)
11. \(\left[r^{n} \cos \omega_{0} n\right] u[n]\)	\(\frac{1-\left[r \cos \omega_{0}\right] z^{-1}}{1-\left[2 r \cos \omega_{0}\right] z^{-1}+r^{2} z^{-2}}\)	\(\|z\|>r\)
12. \(\left[r^{n} \sin \omega_{0} n\right] u[n]\)	\(\frac{\left[r \sin \omega_{0}\right] z^{-1}}{1-\left[2 r \cos \omega_{0}\right] z^{-1}+r^{2} z^{-2}}\)	\(\|z\|>r\)