Z – Transform and Inverse Z – Transform

Contents

**Describe ROC of Z-Transform and its Properties** 2

**Describe Properties of Z-Transform** 3

**Calculate Z-transform of Common Signals** 4

**Describe Inverse Z-transform and Calculate Inverse Z-transform using Partial fraction expansion** 5

**Calculate Inverse Z-transform using Power Series expansion** 6

**Calculate Inverse Z-transform using Contour Integration** 7

**Calculate Inverse Z-transform using Residue theorem** 8

**Describe Unilateral Z-Transform** 9

**Describe Initial-value Theorem and Final-value Theorem** 9

**Find the solution of Difference Equation using Z-Transform** 10

**Explain Transient-State Response and Steady-State Response** 11

**Explain Z-Domain Analysis of Discrete-Time LTI System** 12

**Calculate Impulse Response using Z-Transform** 13

**Calculate Total Response and Transfer Function using Z-Transform** 14

**Define Z-Transform**

The Z-transform is a mathematical tool used in the field of discrete-time signal processing and system analysis. It is the discrete-time counterpart of the Laplace transform, which is used for continuous-time signals and systems.

The Z-transform converts a discrete-time signal or sequence into a complex function of a complex variable “z”. It provides a way to analyze and manipulate discrete-time signals and systems in the frequency domain.

The general form of the Z-transform for a discrete-time signal x[n] is given by:

X(z) = Z{x[n]} = Σ{x[n] * z^(-n)}

where n is the discrete-time index, z is a complex variable, and X(z) represents the Z-transform of the sequence x[n].

The Z-transform has several properties that make it useful for signal analysis and system representation. These properties include linearity, time-shifting, scaling, convolution, and frequency shifting.

The Z-transform is commonly used for various applications, including digital filter design, system modeling and analysis, discrete-time signal processing, and control systems. It provides a powerful tool for studying and manipulating discrete-time signals and systems in both the time and frequency domains.

**Describe ROC of Z-Transform and its Properties**

The Region of Convergence (ROC) is an important concept in Z-transform analysis. It defines the region in the complex plane where the Z-transform converges, meaning the Z-transform is well-defined and finite.

The ROC of a Z-transform can be described by specifying the values of the complex variable “z” for which the Z-transform converges. The ROC is typically specified in terms of the radius of convergence and the angle of convergence.

Properties of ROC:

- Causal ROC: For a causal sequence, the ROC includes the region exterior to the outermost pole in the z-plane.
- Right-Sided ROC: For a right-sided sequence, the ROC includes the region exterior to the outermost pole on the right-hand side of the z-plane.
- Left-Sided ROC: For a left-sided sequence, the ROC includes the region exterior to the outermost pole on the left-hand side of the z-plane.
- Two-Sided ROC: For a two-sided sequence, the ROC includes the region between two poles in the z-plane.

Properties of Z-Transform ROC:

- Causal System: The ROC of the Z-transform of a causal system includes the region exterior to the outermost pole.
- Stable System: A stable system has a ROC that includes the unit circle in the z-plane.
- Finite-Duration Signal: The ROC of the Z-transform of a finite-duration signal is the entire z-plane except possibly at z = infinity.
- Right-Sided Signal: The ROC of the Z-transform of a right-sided signal includes the region exterior to the outermost pole on the right-hand side of the z-plane.
- Left-Sided Signal: The ROC of the Z-transform of a left-sided signal includes the region exterior to the outermost pole on the left-hand side of the z-plane.
- Causal and Right-Sided Signal: The ROC of the Z-transform of a causal and right-sided signal includes the region between the outermost right-sided pole and infinity.

These properties help in determining the convergence and stability of the Z-transform and provide insights into the behavior of the corresponding discrete-time signal or system.

**Describe Properties of Z-Transform**

The Z-transform has several important properties that are useful for analysing discrete-time signals and systems. Here are some of the key properties:

- Linearity: The Z-transform is a linear operator, which means that it satisfies the superposition principle. That is, if x1[n] and x2[n] are two discrete-time signals, and a and b are two constants, then the Z-transform of the linear combination ax1[n] + bx2[n] is given by aX1(z) + bX2(z), where X1(z) and X2(z) are the Z-transforms of x1[n] and x2[n], respectively.
- Time-shifting: If x[n] is a discrete-time signal, and x[n-k] is it’s time-shifted version, then the Z-transform of x[n-k] is given by z^(-k)X(z).
- Convolution: The Z-transform of the convolution of two discrete-time signals x[n] and h[n] is equal to the product of their Z-transforms: X(z)H(z).
- Initial value theorem: The initial value theorem states that the Z-transform of a discrete-time signal x[n] evaluated at z=1 is equal to the signal’s initial value x[0]: lim(z->1) (1-z^(-1))X(z) = x[0].
- Final value theorem: The final value theorem states that the Z-transform of a discrete-time signal x[n] evaluated at z=1 is equal to the signal’s final value, provided that the ROC includes z=1 and the system is stable: lim(n->infinity) x[n] = lim(z->1) (z-1)X(z).
- Multiplication by a complex exponential: If x[n] is a discrete-time signal and a is a complex constant, then the Z-transform of a^n x[n] is given by X(a*z).
- Differentiation in time: The Z-transform of the first difference of a discrete-time signal x[n] is equal to (1-z^(-1))X(z), where X(z) is the Z-transform of x[n].
- Integration in time: The Z-transform of the cumulative sum of a discrete-time signal x[n] is equal to (1/(1-z^(-1)))X(z), where X(z) is the Z-transform of x[n].

In summary, the Z-transform has several useful properties, including linearity, time-shifting, convolution, initial value and final value theorems, multiplication by complex exponentials, differentiation and integration in time. These properties can be used to simplify and manipulate Z-transforms of discrete-time signals and systems.

**Calculate Z-transform of Common Signals**

Here are the Z-transforms of some common signals:

- Unit step function u[n]: The Z-transform of the unit step function is given by:

U(z) = 1/(1-z^(-1)) - Impulse function δ[n]: The Z-transform of the impulse function is given by:

δ(z) = 1 - Exponential function a^n u[n]: The Z-transform of the exponential function a^n u[n] is given by:

X(z) = 1/(1-az^(-1))

This is valid for |z| > |a|. - Sinusoidal function sin(ωn): The Z-transform of the sinusoidal function sin(ωn) is given by:

X(z) = (z sin(ω) – sin(ω))/(z^2 – 2z cos(ω) + 1)

This is valid for |z| > 1. - Sinusoidal function cos(ωn): The Z-transform of the sinusoidal function cos(ωn) is given by:

X(z) = (z – cos(ω))/(z^2 – 2z cos(ω) + 1)

This is valid for |z| > 1. - Geometric sequence r^n u[n]: The Z-transform of the geometric sequence r^n u[n] is given by:

X(z) = 1/(1-rz^(-1))

This is valid for |z| > |r|. - Rectangular pulse function p[n]: The Z-transform of the rectangular pulse function p[n] is given by:

X(z) = (1-z^(-N))/(1-z^(-1))

This is valid for |z| > 1. - Triangular pulse function t[n]: The Z-transform of the triangular pulse function t[n] is given by:

X(z) = (1-z^(-N))/(1-z^(-1))^2

This is valid for |z| > 1.

**Describe Inverse Z-transform and Calculate Inverse Z-transform using Partial fraction expansion**

The inverse Z-transform is a mathematical operation that allows us to convert a Z-transform function back into its original discrete-time domain representation. It is denoted as Z^(-1).

One method to calculate the inverse Z-transform is through partial fraction expansion. The steps involved are as follows:

- Start with the given Z-transform function in the form X(z).
- Factorize the denominator of X(z) into linear and quadratic factors.
- Write X(z) as a partial fraction of these factors, using unknown coefficients A, B, C, etc.
- Express X(z) in terms of these unknown coefficients: X(z) = A/(z – z1) + B/(z – z2) + C/(z – z3) + …
- Determine the values of the unknown coefficients by using algebraic methods such as equating coefficients or manipulating the equation.
- Once the unknown coefficients are determined, write the inverse Z-transform function in the form x[n] = A1 * r1^n + A2 * r2^n + A3 * r3^n + …, where A1, A2, A3, etc., are constants and r1, r2, r3, etc., are the roots of the denominator factors.
- Simplify the expression further if possible, using algebraic manipulations or known Z-transform properties.
- Finally, write the inverse Z-transform function in terms of the discrete-time domain variable n, yielding the original time-domain representation of the function.

It is important to note that the process of calculating the inverse Z-transform using partial fraction expansion can become complex for higher-order polynomials or when dealing with complex roots. In such cases, advanced mathematical techniques like residue theory or contour integration may be employed.

Overall, the inverse Z-transform allows us to recover the original discrete-time signal or system representation from its Z-transform, enabling us to analyze and understand the behavior in the time domain.

**Calculate Inverse Z-transform using Power Series expansion**

To calculate the inverse Z-transform using the power series expansion method, we can express the Z-transform function as a power series and then identify the coefficients of the terms in the series. The steps involved are as follows:

- Start with the given Z-transform function X(z).
- Write X(z) as a power series expansion using the formula:

X(z) = X(0) + X(1)z^(-1) + X(2)z^(-2) + X(3)z^(-3) + …

where X(0), X(1), X(2), X(3), etc., are the coefficients of the power series. - Determine the values of the coefficients X(0), X(1), X(2), X(3), etc. This can be done by multiplying both sides of the equation by appropriate powers of z and collecting coefficients.
- Once the coefficients are determined, write the inverse Z-transform function in the form x[n] = X(0) * δ[n] + X(1) * δ[n-1] + X(2) * δ[n-2] + X(3) * δ[n-3] + …, where δ[n] is the discrete-time impulse function.
- Simplify the expression further if possible, using known properties of the Z-transform or algebraic manipulations.
- Finally, write the inverse Z-transform function in terms of the discrete-time domain variable n, yielding the original time-domain representation of the function.

It is important to note that the power series expansion method may not always be applicable or may lead to a cumbersome calculation for complex Z-transform functions. In such cases, other methods like partial fraction expansion or contour integration may be used to calculate the inverse Z-transform.

Remember to pay attention to the region of convergence (ROC) of the Z-transform, as it determines the range of values for which the inverse Z-transform is valid.

Overall, the power series expansion method provides a systematic approach to calculate the inverse Z-transform and obtain the time-domain representation of a discrete-time signal or system.

**Calculate Inverse Z-transform using Contour Integration **

To calculate the inverse Z-transform using contour integration, we can use the Residue theorem. The steps involved are as follows:

- Start with the given Z-transform function X(z) and determine its region of convergence (ROC).
- Identify the poles of X(z), which are the values of z for which X(z) becomes infinite.
- Choose a contour in the complex plane that encloses all the poles of X(z) within it.
- Apply the Residue theorem, which states that the inverse Z-transform can be obtained by integrating X(z) over the chosen contour and summing the residues of the poles enclosed by the contour.
- Evaluate the residues of the poles. The residue of a pole at z = z0 can be calculated as Res[z0] = lim(z->z0) [(z – z0) * X(z)].
- Integrate X(z) along the chosen contour, taking into account the direction of integration and any branch cuts or singularities.
- Once the contour integration is performed, simplify the resulting expression to obtain the inverse Z-transform.
- Express the inverse Z-transform in terms of the discrete-time domain variable n, yielding the time-domain representation of the function.

It is important to note that contour integration requires careful consideration of the ROC and the behavior of X(z) in the complex plane. The contour should be chosen such that it does not pass through any singularities or branch cuts that may affect the integral.

Contour integration is a powerful method for calculating the inverse Z-transform, especially for Z-transform functions with complex poles or when other methods such as partial fraction expansion or power series expansion may not be feasible or efficient.

**Calculate Inverse Z-transform using Residue theorem**

To calculate the inverse Z-transform using the residue theorem, follow these steps:

- Start with the given Z-transform function X(z) and determine its region of convergence (ROC).
- Identify the poles of X(z), which are the values of z for which X(z) becomes infinite.
- Compute the residues of the poles. The residue of a pole at z = z0 can be calculated as Res[z0] = lim
_{(z->z0)}[(z – z0) * X(z)]. - Determine the ROC of the inverse Z-transform based on the locations of the poles. The ROC will be the complement of the union of the ROCs of X(z) and the poles.
- Write the inverse Z-transform as a sum of terms, where each term corresponds to a residue and a corresponding pole. The inverse Z-transform can be expressed as X[n] = (1/2πj) ∮ X(z) * z^(n-1) dz, where the integration is performed along a closed contour in the ROC.
- Evaluate the inverse Z-transform by calculating the contour integral. The contour integral can be evaluated by summing the residues of the poles enclosed by the contour.
- Simplify the resulting expression to obtain the inverse Z-transform in terms of the discrete-time domain variable n.

It’s important to note that the residue theorem is applicable only when the ROC includes a circular contour that encloses all the poles of X(z) and no other singularities. In cases where the ROC does not satisfy these conditions, additional methods such as partial fraction expansion or power series expansion may be needed to calculate the inverse Z-transform.

**Describe Unilateral Z-Transform**

The unilateral Z-transform is a mathematical technique that is used to analyse discrete-time signals and systems. It is similar to the unilateral Laplace transform, which is used to analyse continuous-time signals and systems.

The unilateral Z-transform is defined as:

X(z) = Z{x[n]} = x[n] z^{{-n}}

where x[n] is the discrete-time signal and z is the complex variable. The range of z for which the above series converges is called the region of convergence (ROC) of the Z-transform.

Unlike the bilateral Z-transform, which considers both positive and negative values of n, the unilateral Z-transform only considers non-negative values of n. This means that the unilateral Z-transform is a one-sided transform, and it is useful for analysing causal systems, where the output of the system depends only on past and present inputs.

The inverse unilateral Z-transform is used to recover the original signal x[n] from its Z-transform X(z). There are several methods to calculate the inverse unilateral Z-transform, including partial fraction expansion, power series expansion, and contour integration. The choice of method depends on the complexity of the Z-transform and the desired accuracy of the inverse transform.

**Describe Initial-value Theorem and Final-value Theorem**

The initial-value theorem and final-value theorem are two important properties of the unilateral Z-transform, which are similar to the initial-value theorem and final-value theorem of the unilateral Laplace transform.

The initial-value theorem states that the initial value of a discrete-time signal x[n] can be calculated from its Z-transform X(z) as:

x[0] = lim_{{z→∞}} X(z)

This means that the value of x[0] can be obtained by taking the limit of X(z) as z approaches infinity.

The final-value theorem, on the other hand, states that the final value of a discrete-time signal x[n] can be calculated from its Z-transform X(z) as:

x[∞] = lim_{{z→1}} (z-1) X(z)

This means that the value of x[∞] can be obtained by taking the limit of (z-1)X(z) as z approaches 1. Note that this theorem only applies to signals that are bounded and stable.

Both the initial-value theorem and final-value theorem are useful for the behaviour of discrete-time signals and systems, particularly in the time domain. They provide a way to calculate the initial and final values of a signal without having to perform an inverse Z-transform.

**Find the solution of Difference Equation using Z-Transform **

To find the solution of a linear difference equation using the Z-transform, you can follow these steps:

- Take the given difference equation and express it in the Z-domain by applying the Z-transform to both sides of the equation. Replace the time-domain variables with their corresponding Z-transform counterparts.
- Solve for the Z-transform of the desired output sequence or variable. This is usually denoted as Y(z).
- Manipulate the equation to isolate Y(z) on one side and obtain an expression for Y(z) in terms of the Z-transform of the input sequence or variable, X(z), and any other known Z-transforms.
- Take the inverse Z-transform of Y(z) to obtain the time-domain solution of the difference equation. This will give you the desired output sequence or variable as a function of time.

The exact method and steps for solving the difference equation using the Z-transform may vary depending on the specific form of the equation and the initial or boundary conditions. It often involves algebraic manipulation, partial fraction expansion, and the use of Z-transform properties.

It’s important to note that the Z-transform assumes a particular region of convergence (ROC) for which the series converges. The choice of ROC affects the validity and interpretation of the solution. Therefore, when using the Z-transform to solve a difference equation, it’s crucial to ensure that the chosen ROC corresponds to the region of interest and satisfies the convergence conditions.

**Explain Transient-State Response and Steady-State Response**

Transient-state response and steady-state response are two components of the overall response of a system to an input signal. They describe different aspects of how the system behaves over time.

- Transient-State Response:
- The transient-state response refers to the behavior of the system immediately after a change in the input signal or when the system is initially energized.
- It represents the time-varying part of the system’s response, which gradually settles down and approaches a steady-state.
- During the transient period, the system’s output may exhibit oscillations, overshoot, or undershoot before eventually stabilizing.
- The transient response provides information about the system’s dynamic behavior, including its time constants, damping, and initial conditions.

- Steady-State Response:
- The steady-state response refers to the behavior of the system once it has reached a stable operating condition and all transient effects have decayed.
- It represents the long-term, time-invariant part of the system’s response.
- In the steady-state, the system’s output remains constant or follows a predictable pattern in response to a periodic input signal.
- The steady-state response is characterized by parameters such as gain, phase shift, and frequency response, which describe how the system affects different frequencies in the input signal.
- It provides information about the system’s ability to accurately reproduce and amplify the input signal over time.

In summary, the transient-state response describes the system’s behavior during the initial phase or following a change in the input signal, while the steady-state response represents the system’s behavior once it has settled into a stable operating condition. Understanding both components is essential for analyzing and designing systems in various fields, including control systems, signal processing, and communications.

**Explain Z-Domain Analysis of Discrete-Time LTI System**

Z-domain analysis is a method used to analyze discrete-time linear time-invariant (LTI) systems in the frequency domain. It involves representing the system’s input, output, and transfer function in the Z-domain using the Z-transform. This analysis technique provides insights into the system’s frequency response, stability, and transient behavior.

The Z-transform is a mathematical tool that converts a discrete-time signal or system from the time domain to the Z-domain. It is analogous to the Laplace transform used in continuous-time systems. By applying the Z-transform to the difference equation representing the discrete-time LTI system, we can obtain the system’s transfer function in the Z-domain.

The Z-transform of a discrete-time signal x[n] is denoted as X(z), where z is a complex variable. Similarly, the Z-transform of the system’s output y[n] is denoted as Y(z). The transfer function H(z) represents the relationship between the input and output signals in the Z-domain and is defined as the ratio of Y(z) to X(z).

Z-domain analysis enables us to perform various operations such as multiplication, addition, and differentiation in the frequency domain. This allows us to determine important characteristics of the system, including frequency response, stability, and transient behavior.

Some key aspects of Z-domain analysis include:

- Frequency Response: By evaluating the transfer function H(z) at different values of z, we can determine the system’s frequency response. This provides information about how the system responds to different frequencies in the input signal, including gain, phase shift, and frequency selectivity.
- Pole-Zero Analysis: The zeros and poles of the transfer function H(z) in the Z-domain provide insights into the system’s stability and transient response. The locations of the poles determine the stability of the system, and the presence of zeros affects the system’s frequency response and transient behavior.
- System Identification: Z-domain analysis can be used to identify the system’s parameters, such as the coefficients in the difference equation, based on the input-output relationship. This is particularly useful in system modeling and control design.
- Inverse Z-Transform: After performing analysis in the Z-domain, the inverse Z-transform can be applied to obtain the time-domain representation of the system’s output. This allows us to examine the system’s behavior in the time domain.

Z-domain analysis is widely used in digital signal processing, control systems, and communication systems. It provides a powerful framework for analyzing and designing discrete-time LTI systems and understanding their behavior in the frequency domain.

**Calculate Impulse Response using Z-Transform **

To calculate the impulse response of a discrete-time LTI system using the Z-transform, you need to follow these steps:

- Start with the transfer function of the system in the Z-domain. Let’s denote it as H(z).
- Find the inverse Z-transform of H(z) to obtain the corresponding difference equation. This equation represents the system’s input-output relationship in the time domain.
- Identify the coefficients of the difference equation. Let’s assume the difference equation is of the form:

y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + … – a1y[n-1] – a2y[n-2] – …

Here, the coefficients b0, b1, b2, … are the feedforward coefficients, and a1, a2, … are the feedback coefficients. - Set the initial conditions to zero. Since we are calculating the impulse response, the initial conditions are assumed to be zero, i.e., y[-1] = y[-2] = … = 0.
- Determine the values of the impulse response sequence h[n]. The impulse response is the output of the system when the input is an impulse, which is represented by x[n] = δ[n], where δ[n] is the discrete-time delta function.

By substituting x[n] = δ[n] into the difference equation and considering the initial conditions, you can solve for the values of h[n]. - Once you have determined the values of h[n], you have obtained the impulse response of the system. The sequence h[n] represents the system’s output when the input is an impulse.

It’s important to note that the complexity of calculating the impulse response using the Z-transform depends on the transfer function H(z) and the nature of the difference equation. In some cases, it may be straightforward to directly obtain the impulse response, while in other cases, additional algebraic manipulations or partial fraction decomposition may be required.

Overall, the Z-transform provides a powerful tool for analyzing discrete-time systems, and by calculating the impulse response, you can gain insights into the system’s behavior and characteristics.

**Calculate Total Response and Transfer Function using Z-Transform**

To calculate the total response and transfer function of a discrete-time LTI system using the Z-transform, you need to follow these steps:

- Start with the difference equation that represents the system’s input-output relationship in the time domain. Assume the difference equation is of the form:

y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + … – a1y[n-1] – a2y[n-2] – …

Here, the coefficients b0, b1, b2, … are the feedforward coefficients, and a1, a2, … are the feedback coefficients. - Take the Z-transform of both sides of the difference equation. Let Y(z) and X(z) represent the Z-transforms of y[n] and x[n], respectively. Apply the Z-transform property of linearity and the Z-transform properties of the shifting theorem to obtain:

Y(z) = H(z)X(z),

where H(z) is the transfer function of the system. - Solve for the transfer function H(z) by rearranging the equation:

H(z) = Y(z) / X(z).

Here, H(z) represents the system’s transfer function, which relates the Z-transforms of the output and input. - To calculate the total response y[n], you need to find the inverse Z-transform of Y(z). Use techniques such as partial fraction expansion, power series expansion, or residue theorem to obtain the inverse Z-transform of Y(z) and determine the time-domain expression for y[n].

The total response y[n] represents the output of the system in the time domain given a specific input x[n].

It’s important to note that the complexity of calculating the total response and transfer function using the Z-transform depends on the nature of the difference equation and the transfer function. In some cases, it may be straightforward to directly obtain the transfer function and inverse Z-transform, while in other cases, additional algebraic manipulations or techniques may be required.

By using the Z-transform, you can analyze and characterize the behavior of discrete-time LTI systems in the frequency domain and obtain their total response and transfer function. These results provide valuable insights into the system’s dynamics and can be used for system design and analysis.