# EE445S Real-Time Digital Signal Processing Laboratory - Midterm #1

Prof. Brian L. Evans

Midterm #1 will be an open book, open notes exam scheduled to last the entire period. Midterm #1 questions will come from lecture and lab. It is possible that one problem on the midterm may require you to write TMS320C6700 C/assembly code. TMS320C6700 assembly language instructions are tabulated in the slides for lectures 2 and 9.

Here are several example midterm #1 exams:

- Spring 2014 without solutions and with solutions
- Fall 2013 without solutions and with solutions
- Spring 2013 without solutions and with solutions

Many past midterm #1 exams (mostly with solutions) are available in the course reader.

For Midterm #1, you will be responsible for the material in

1. In-class lectures 0-6 and lecture 9
2. In-class demonstrations, including those from DSP First. Username and password information.
3. In-class discussions
4. Johnson, Sethares and Klein, Software Receiver Design, chapters 1-7 and appendices A & G
5. Welch, Wright and Morrow, Real-Time Digital Signal Processing, chapters 1-7 and appendices A-D
6. Laboratory assignments 1-3
7. Handouts in the course reader entitled Convolution Example, Fundamental Theorem of Linear Systems, Modulation Example, Sample Quizzes, Tapped Delay Line on C6700 DSP, and All-pass Filters
8. Homework assignments 0-3 and their solution sets

For Midterm #1, you will be responsible for the following topics:

1. Communication system introduction (Lecture 0)
2. Conventional programmable digital signal processor architectures (Lecture 1)
3. Bandwidth (Lecture 2)
4. Sinusoidal generation (Lecture 2)
5. Upconversion and downconversion (Lectures 2 and 4)
6. Continuous-time system properties (Lecture 3)
7. Basic continuous-time signals (Lecture 3)
8. Discrete-time system properties (Lecture 3)
9. Basic discrete-time signals (Lecture 3 and 5)
10. Fundamental Theorem of Linear Systems for continuous-time systems (Handout F)
11. Sampling theorem (Lecture 4)
12. Sampling and aliasing (Lecture 4
13. Bandpass sampling (Lecture 4)
14. Discrete-to-continuous conversion (Lecture 4)
15. Fundamental Theorem of Linear Systems for discrete-time systems (Lecture 5)
16. Z-transforms (Lecture 5 and 6; Johnson & Sethares Appendix A.4)
17. Transfer functions (Lecture 5 and 6; Johnson & Sethares 4.5)
18. Relationships between transforms (Lecture 5)
19. Digital FIR filter implementation (Lecture 2 and 5)
20. Digital FIR filter analysis (Lecture 5)
21. Stability of continuous-time and discrete-time filters (Lecture 5 and 6)
22. Digital IIR filter design by pole-zero placement (Lecture 6)
23. Three direct form IIR filter structures (Lecture 6)
24. Classical IIR filter design methods (Lecture 6)
25. Implementing IIR filters in cascade of biquads based on quality factors (Lecture 6)
26. Modern programmable digital signal processor architectures (Lecture 9)

Here are the key parts of Lathi's Linear Systems and Signals (second edition) and Roberts' Signals and Systems and Oppenheim and Willsky's Signals and Systems (second edition)

Oppenheim & Willsky (2nd ed) | Roberts | Lathi (2nd ed) | Topics
1.6 | 1.2 | 1.7 | System properties
1.3-1.4 | 2.3 | 1.4 | Basic continuous-time signals
3.2 | 2.4 | 2.4-4 | Fundamental Theorem of Linear Systems for continuous-time systems **
1.3-1.4 | 3.4 | 3.3 | Basic discrete-time signals
3.2 | 3.5 | 3.8-3 | Fundamental Theorem of Linear Systems for discrete-time systems **
9.7.2 | 2.6 | 2.6 | Stability of continuous-time filters
10.7.2 | 3.6 | 3.10 | Stability of discrete-time filters
10.1-10.3 | 5.1 | 5.1 | Z transforms
10.5 | 5.2 | 5.2 | Properties of the z-transform
10.7.3-10.7.4 | 5.3 | 5.3 | Transfer functions, and filter design by pole-zero placement
| Ex. 5.13 |  | Notch filters
| Prob. 5.5-7 |  | All-pass filters
10.8 | 5.4 | 5.4 | Realizations of transfer functions
4.3-4.4 | 7.4 | 7.3 | Fourier transform properties (esp. the frequency shifting property for sinusoidal amplitude modulation)
| 7.10 | ?.?? | Pulse shapes
7.1 | 8.1 | 8.1 | Sampling theorem
7.4 | 8.2 | 8.4 | Spectral analysis of sampling theorem

** Please see Appendix F and slide 5-13 in the course reader for the fundamental theorem. Oppenheim & Willsky (2nd ed) covers a slightly different version of the fundamental theorem, in which a complex exponential is the input to a linear time-invariant system. Lathi also has that version as well.

Here are examples of calculations that many students have had trouble getting right on midterm #1.

The first common mistake is deriving transfer functions. One source is Lathi's Linear Systems and Signals book, which uses the operator notation D y(t) to represent (d/dt) y(t) or y'(t). This is not an error. Lathi's book continues as follows, given input signal x(t) and output signal y(t):

    y'(t) + y(t) = x(t)
    D y(t) + y(t) = x(t)

So far, so good. The problem starts when Lathi's book does the following:

    (D + 1) y(t) = x(t)

From the above Lathism, many students erroneously conclude that

    y(t)     1
    ---- = -----
    x(t)   D + 1
           y(t)
    h(t) = ----
           x(t)

where h(t) is the transfer function of the system in the time domain.

Comments:

- Clearly, h(t) = y(t) / x(t) is not a meaningful way to represent a system, and moreover, it isn't valid for values of t when x(t) is zero.
- h(t) should have been the impulse responses of an LTI system-- a differential equation with initial conditions is only an LTI system if the initial conditions are zero. In that case,

      y(t) = h(t) * x(t)

  where * means convolution.
- Transfer functions are derived in either the generalized transform domain (i.e. Laplace or z) or the Fourier domain. The transfer function in the Fourier domain is also known as the frequency response. There is no concept of a transfer function in the time domain.

Updated 10/13/14.