The final is May 11, 1999  - 12:20 ­ 2:10 PM in BAC. You should basically study both from your notes AND from the relevant sections in Oppenheim and Schafer

Sections to Study:

All of Chapter 2

3.1, 3.2, 3.3, 3.4,

5.1, 5.2, 5.2.1, 5.3, 5.3.1 5.3.2, 5.7, 5.7.1-3

6.1, 6.2

7.1, 7.1.1, 7.1.2, 7.1.3, 7.2, 7.2.1, 7.2.2, 7.2.3, 7.3

8.1, 8.2, 8.3, 8.4, 8.5, 8.6

9, 9.3, 9.4,

10.1, 10.2, 10.4

See also relevant sections of Appendix A,B, and C as needed

Material for examination by topic:

 -FIR and IIR Discrete-time Linear Systems: difference equations, convolution sum, unit impulse, closed-form expressions for transient and steady-state responses, stability, causality, time invariance, frequency-response function, impulse response, sinusoidal steady-state response

-The Discrete-time Fourier Transform (DTFT): properties and their applications, fundamental transform DTFT pairs (unit impulse, sample sinusoids, digital sinc, exponential sequences), application to system and signal analysis

-Z-transform: Definition, ROC and its properties, right- (causal) and left- (anticausal) sequences, relation to the DTFT, properties of the z-transform, fundamental z-transform pairs (unit impulse, exponential sequences, sinusoids, unit step sequence), transfer function and its relation to the impulse response and the frequency response, poles and zeros, inverse z-transform and its application to the recovery of causal and anticausal sequences from their z-domain functions, residues and their application to the recovery of causal sequences from their z-transform, impulse, transient- and steady-state responses of linear discrete-time systems, simulation diagrams for FIR and IIR filters

-Poles and Zeros and their effect on the magnitude frequency response, design by pole-zero placement, linear phase FIR filters, group delay, symmetries

-The DFT, definitions, properties and their applications, spectral leakage, windows, linear and periodic convolution, relations to the DTFT, response to sinusoids, digital sinc, periodicity, resolution

-The FFT and its relation to the DFT, applications to signal analysis and data compression (e.g., concepts addressed in your project), Decimation In Time FFT, complexity issues

-Design of linear phase filters using the Fourier series and sampled windows. The impulse invariance method. The bilinear transformation and its application to analog filter approximations. Butterworth filter design.

-Random signal processing, stationarity and ergodicity, definitions of first- (mean) and second- (autocorrelation) order statistics, power spectral density, response of linear systems to random signals

Test Structure:

There will be two parts. The first part will consist of short questions (perhaps some multiple choice and true/false as well). No partial credit on those. This part will account for about 40% of the final grade. The second part will have problems and there will be partial credit on these. This part will account for about 60% of the final grade. About 50-60% of the short questions and problems will be on the most recent material after test 2 (Butterworth, DFT/FFT, random signals)

The final test is comprehensive and closed book/closed notes. You are allowed, however, to prepare and carry one 8.5x11 sheet (both sides) with formulas, etc.

Objectives:

You should be able to:

- write difference equations from simulation diagrams and vice versa.

- Find impulse responses for FIR and IIR systems

- Find sinusoidal steady-state responses for FIR and IIR systems

- Identify linearity, time invariance, and causality

- Test for stability

- Evaluate (and sketch) Fourier spectra of continuous and sampled signals

- demonstrate knowledge of sampling theorem

- demonstrate knowledge of properties of the DTFT, and z-transform (those properties covered in class), Application and proofs

- Evaluate inverse z transforms by inspection (left- and right- handed sequences), partial fraction (left- and right- handed sequences), and residue analysis (for right- handed sequences), apply z-transform to system analysis

- Write transfer functions from simulation diagrams and from difference equations and vice versa, Identify poles of the system, write transient- and steady-state responses of the system given the system parameters and the input

- Give pole-zero diagrams, design simple filters by pole-zero placement, realize filters from pole-zero locations, sketch magnitude frequency response from pole zero locations

- identify symmetries for linear phase FIR design, show that symmetries and anti-symmetries produce piecewise linear phase, linear phase filter design

- write down definitions and properties of the DFT and IDFT, know how to apply the properties of the DFT, Proofs of linearity, time/frequency shift, time convolution, Parsevals' theorem, identify and exploit symmetries in the time- and frequency- domain, determine simple DFT transform pairs using the definition and properties

- know frequency resolution issues and complexity of FFT algorithms, utility of windows, leakage, etc

- know the concepts associated with the derivation of the DIT/FFT and DIF/FFT algorithms

- know how to design FIR linear-phase filters using the Fourier series and sampled windows

- know how to design a digital Butterworth filter using the bilinear transformation

- know how to design a simple filters by analog approximation and impulse invariance or bilinear transformation

- know definitions of signal statistics, mean, variance, autocorrelation, spectral density, white noise, colored noise

- given input statistics (mean, variance, autocorrelation) and system parameters you should be able to determine output statistics