EECS451: Digital Signal Processing and Analysis: Course Goals

	Catalog Description:

Introduction to discrete-time (DT) signal processing.
The family of Fourier Transforms including the Discrete Fourier Transform (DFT),
discrete-time Fourier series (DTFS), discrete-time Fourier transform (DTFS),
and fast Fourier transform (FFT).  Signal sampling and reconstruction.
Design and analysis of digital filters.  Correlation and spectral estimation.
Laboratory experiences exercise and illustrate the concepts presented.


	Course Objectives:

1. To teach students the concepts of discrete-time signals, including
mathematical representations, properties, frequency content, and aliasing.
2. To teach students the concepts of linear time-invariant discrete-time
systems, including representations, properties, convolution relationship,
and analysis techniques based on Fourier and Z transforms.
3. To introduce the concepts of filter design.


	Course Outcomes
	(Students should know about and/or be able to do these by end of course)

	[Analytical Outcomes]:

Determine (possibly aliased) digital frequency of a sampled sinusoidal signal.

Convert DT signals between different notation representations.

Identify basic components of a DSP system (A/D, signal processor, D/A)

Identify classes of DT systems:
determine whether a DT system is linear, time-invariant, causal, stable, dynamic

Awareness of the superposition property and summation for linear systems.

Perform analytical convolution of two DT signals,
either in time domain or using Z-transforms, or using Fourier transforms.
Apply to determining response of LTI system to given input signal.

Convert between the following six representations of linear time invariant
discrete time systems represented by constant coefficient difference equations:
block diagram, recursive input-output expression, pole-zero diagram, system
function in factored or expanded form, impulse response, frequency response.

Awareness of the unique role of LTI systems represented by constant-coefficient
difference equations.

Use convolution properties to simplify LTI systems.

Determine Z-transform of DT signal and specify ROC,
using Z-transform properties to solve such problems efficiently.

Invert Z-transform by power-series expansion, table-lookup, and/or PFE.
Apply to: finding impulse response of DT system, convolution/system response.

Identify LTI system properties from system function / pole-zero plot
(transient/steady-state response, causality, stability).

Choose appropriate member of family of Fourier Transforms for a given problem.

Apply the (inverse) DTFS to decompose a DT periodic signal
into complex exponentials.
Apply the DTFS to synthesize a DT periodic signal.

Apply eigenfunction concept to determine response of LTI system
to an eternal periodic input signal.

Determine DTFT of aperiodic DT signals, using FT properties or from Z-transform.
Determine inverse DTFT, using all FT properties.

Recognize Gibb's phenomenon in truncated Fourier series expansions.

Determine spectrum of sampled signal from spectrum of analog signal, given T,
and identified aliased frequency components if any.

Understand origins of Nyquist sampling rate and sinc reconstruction formula
for bandlimited signals.

Determine transient and steady-state response of LTI system to persistent input.

Sketch and/or determine magnitude response from pole-zero plot of a filter.

Identify filter type from pole-zero plot.

Determine invertibility of and inverse filter for LTI systems.

Determine DFT of DT signal directly from formula, or by using properties,
or by sampling DTFT if time-limited, or numerically using FFT.

Relate different members of FT family conceptually and mathematically.

Perform circular convolution of two time-limited DT signals.

Understand how zero-padding allows linear convolution from circular convolution.

Awareness of overlap-add / overlap-sum methods for long data sequences.

Determine spectral resolution provided by DFT, choose N accordingly.

Understand radix-2 derivation of FFT algorithm and its computational savings.

Identify basic tradeoffs in digital filter design.

Design FIR filters using window method or equiripple method.

Design IIR filters by bilinear transformation of analog filters,
understanding frequency warping.  Make rough pole-zero design.

Awareness of basic practical issues in anti-aliasing filters, quantization,
D/A conversion.

Generalize from skills and concepts above to solve related but different
signal processing problems.

	[Computational Outcomes (Matlab Skills)]:

Plot signals and transforms appropriately.

For difference equation systems, create and display:
pole-zero plots, PFE, impulse response, magnitude response.

Convolve signals using conv, fftfilt, filter, or fft/ifft.
Understand which choice to use when.

Understand fftshift().

Implement simple DT algorithms such as FFT of 2 real sequences.
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	Assessment

1. In-class closed book exams test objectives #1-N for individual students.
2. Weekly problem sets test objectives #1-N under less time pressure and with
   the possibility of student collaboration.