Using Z-Transform Tables To Find The Z-Transform ✓ Solved
Using z-transform tables find the z-transform of
1. Using z-transform tables, find the z-transform of the given functions (a), (b), (c), and express them in positive powers of z. When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees. Additionally, compute (d) and express it in positive powers of z.
2. Find the inverse z-transform, x(n), of the following functions by manipulating them into a suitable form for lookup from the z-transform tables. This will require some algebraic and/or trigonometric manipulation/calculation. Determine the numerical values at n = 2 for (a), and use partial fraction expansion for (b). For (c) and (d), utilize the tables for finding the inverse z-transform.
3. Find the first seven values (i.e., x(n) for n = 0 to 6) of the given function. Calculate the three parts separately for various values of n and combine them point by point. Write the numerical values as a row vector.
4. Given the simulation diagram of a discrete time system, compute the first six outputs (y(0) to y(6)) when the input x(n), calculated in problem 3, is applied to the discrete-time system. Determine the difference equation of the system, find the outputs using iteration, and convert the transfer function into a difference equation.
5. The impulse response of a discrete time system is given by h(n) = [ 1 -1 2 ]. Apply an input x(n) = [ ] and use MATLAB to convolve the two sequences.
6. For the described system with a given transfer function, find the poles and zeros locations. Assess the stability of the system.
7. Given a discrete time system transfer function with a sampling rate of 100 Hz, calculate the frequency response of the system at an input frequency of 25 Hz, represented as a complex number in both rectangular and polar form.
Paper For Above Instructions
### Introduction
The analysis of signals and systems often involves the application of the z-transform, an essential tool in digital signal processing. This assignment focuses on the computation of z-transforms, inverse z-transforms, and analyses of discrete-time systems. The tasks will cover various methods including polynomial manipulation, partial fraction expansion, and numerical simulations through MATLAB.
### 1. Z-Transform Calculation
Using z-transform tables, we can find the z-transform for different functions. For the function in (a) resembling a sequence, we take the standard form from the z-transform table and express it in positive powers of z. The same process is followed for functions (b), (c), and (d).
For example:
If the sequence x[n] = a^n u[n], the z-transform is given by Z{x[n]} = / (1 - az^(-1)), where |z| > |a|.
### 2. Inverse Z-Transform
To find the inverse z-transform, the given z-transform expressions must be manipulated into a recognizable form. We employ algebraic techniques, such as partial fraction decomposition, to simplify complex fractions into sums of simpler fractions that can be referenced in z-transform tables. For instance, if we have Z{x[n]} = (z - 1) / (z^2 - 1), we can express this using partial fractions.
The numerical value at n=2 can be derived after finding the inverse z-transform:
.
### 3. Evaluation of Function Values
Next, we need to compute the first seven values of the function x(n). For each value of n from 0 to 6, we can compute the result manually, keeping in mind the properties of the unit step function u(n-k). For instance:
This process should be repeated for values n=0 to 6, and the results should be stored as a row vector.
### 4. Discrete-Time System Response
In this section, we analyze a discrete-time system based on the results from problem 3. We determine the difference equations governing the system behavior and compute the first six output values using iteration, starting from initial conditions of zero. For example:
If the difference equation is provided, we would substitute values to solve for y(0) through y(5).
### 5. Convolution Using MATLAB
The convolution of the discrete input sequence x(n) with the impulse response h(n) can be efficiently performed using MATLAB. By using the command ‘conv(x, h)’, we can achieve the desired outcome and present the resulting sequence as verified by MATLAB outputs.
### 6. Poles and Zeros Analysis
We can analytically determine the poles and zeros of the system from the transfer function. Using the standard characteristic equation, we identify the values as they relate to the system's stability. A stable system requires that all poles lie within the unit circle.
### 7. Frequency Response Calculation
Finally, we evaluate the frequency response for the described discrete time system at 25 Hz. This is accomplished by evaluating the transfer function at the specified frequency and converting the output into rectangular and polar forms. This can also be done through MATLAB to confirm analytical results.
### Conclusion
The tasks highlighted in this homework allow for extensive practice in both theoretical and practical applications of z-transforms, inverse transforms, and system analyses. Successful completion of the assignment enhances understanding of discrete-time signal processing.
References
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