0.3 Lab 3 - frequency analysis

Questions and comments

Introduction

In this experiment,we will use Fourier series and Fourier transforms to analyze continuous-time and discrete-time signals and systems.The Fourier representations of signals involve the decomposition of the signal in terms of complex exponential functions.These decompositions are very important in the analysis of linear time-invariant (LTI) systems, due to the property that theresponse of an LTI system to a complex exponential input is a complex exponentialof the same frequency! Only the amplitude and phase of the input signal are changed.Therefore, studying the frequency response of an LTI system gives complete insight into its behavior.

In this experiment and others to follow, we will use the Simulink extension to Matlab.Simulink is an icon-driven dynamic simulation package that allows the user to represent a systemor a process by a block diagram. Once the representation is completed,Simulink may be used to digitally simulate the behavior of the continuous or discrete-time system.Simulink inputs can be Matlab variables from the workspace, or waveforms or sequences generated by Simulink itself.These Simulink-generated inputs can represent continuous-time or discrete-time sources.The behavior of the simulated system can be monitored using Simulink's version of common lab instruments, such as scopes, spectrum analyzers andnetwork analyzers.

Background exercises

Synthesis of periodic signals

Each signal given below represents one period of a periodic signal with period $T_0$ .

1. Period $T_0=2$ . For $t\in [0,2]$ :
   $$s\left(t\right)=\text{rect}(t-\frac{1}{2})$$
2. Period $T_0=1$ . For $t\in [-\frac{1}{2},\frac{1}{2}]$ :
   $$s\left(t\right)=rect\left(2t\right)-\frac{1}{2}$$

• i Compute the Fourier series expansion in the form
  $$s\left(t\right)=a_0+\sum _{k=1}^{\infty }{A}_{k}sin(2\pi k{f}_{0}t+{\theta }_k)$$
  where $f_0=1/T_0$  .
  Hint You may want to use one of the following references: Sec. 4.1 of ``Digital Signal Processing'', by Proakis and Manolakis, 1996;Sec. 4.2 of ``Signals and Systems'', by Oppenheim and Willsky, 1983; Sec. 3.3 of ``Signals and Systems'', Oppenheim and Willsky, 1997.Note that in the expression above, the function in the summation is $sin(2\pi k{f}_{0}t+{\theta }_k)$ , rather than a complex sinusoid. The formulas in the above referencesmust be modified to accommodate this. You can compute the cos/sinversion of the Fourier series, then convert the coefficients.
• ii Sketch the signal on the interval $[0,T_0]$ .

Magnitude and phase of discrete-time systems

For the discrete-time system described by the following difference equation,

• i Compute the impulse response.
• ii Draw a system diagram.
• iii Take the Z-transform of the difference equation using the linearity and the time shifting properties of the Z-transform.
• iv Find the transfer function, defined as
  $H\left(z\right)\equiv \frac{Y\left(z\right)}{X\left(z\right)}$
• v Use Matlab to compute and plot the magnitude and phase responses, $\left|H\right(e^{j\omega }\left)\right|$ and $\angle H\left(e^{j\omega }\right)$ , for $-\pi <\omega <\pi$    . You may use Matlab commands phase and abs .