0. Motivation (Why do you care?)
A. The Fourier Transform
1. One of the three things that defines you as an EE
or CSE (say, relative to a CS major - work with
*great* CS post-docs in networking who do not get it)
2. Functionally
a. Most things are linear (or we pretend they are)
b. Convolution
(i) the formula - how's it look?
(ii) pain
+ it is hard for you
+ it is even hard for the computer
(iii) just as importantly: it is non-intuitive!
3. Application
a. Communications Engineers: up, down, IF
(i) talk about the cell phone (radio in general)
(ii) what are you doing on your radio dial
(iii) 1 GHz = 1 foot
b. Microwaves:
(i) noise spectrum, filtering
(ii) anybody seen a spectrum analyzer while on co-op?
B. Discrete-time
1. Welcome to the 21st (and late 20th) century: we use digital
processing [e.g. cell phone again]
2. Reminder of what that means:
a. Analog signal (voltage on a wire) sampled and quantized -
numbers in a memory, represented by bits
b. A program operates on these numbers to create more numbers
in memory
c. Numbers are converted back to an analog signal (voltage on
a wire)
3. That sounds like a lot of work. Why would you do that?
a. Reliable, reproducible (exact versus tolerances)
b. Flexible
4. Applications: this is CSE
a. Team Wolf application
b. Graffiti helicopter application
c. 1/N \sum_{i=0}^{N-1} a_i:
(i) this is a digital filter
(ii) get this and you will be king
I. Signals (by far the dullest part of the course)
A. Review - complex numbers (Lathi, B.1)
1. a + bj (add, subtract, multiple, divide)
2. r e^{j \theta} and the unit circle
3. Magnitude, magnitude, magnitude
a. Properties
(i) What are they?
(ii) What if you forget?
b. Example: H(f)
B. Basic Time-domain Signals
1. Definition of x(t) (and voltage on a wire)
2. Operations - pictures, pictures, pictures (Lathi, 1.2)
a. Time shifting (hard): x(t-1)
b. Amplitude shifting (easy): 1+x(t)
b. Scaling in time (hard): x(2t)
c. Scaling in amplitude (easy): 2 x(t)
d. Time reversal: x(-t)
e. Time shifting, reversal, and scaling together x(-2t-2)
3. Properties
a. Energy and power (Lathi, 1.1)
b. Periodic and aperiodic (Lathi, 1.3-3)
4. Some useful signals (Lathi, 1.4)
a. Unit step u(t)
b. Unit impulse \delta(t)
II. Continuous-Time Systems
A. The Box - how do you characterize?
1. Table of every input and output...ouch
2. Is there another way? Others have shown the way if you can
prove a couple of key things:
a. linearity (akin to superposition)
b. time-invariance (it acts the same all of the time)
B. Properties of Systems
1. Linear and nonlinear
a. Definition of linearity
b. Example 1: how do you prove
c. Can you give me an example of something that is nonlinear?
(i) mathematically: y(t) = x^2(t)
+ counterexample inputs
+ why the proof would break down
(ii) real-world?
+[something from ECE 323 - diode?]
+ power amplifiers - why it is such a big deal in
wireless communications
d. What do you do if it not linear? Try to break into regions
where it is linear and work there; in other words, linearity
is a *big* deal.
2. Time-invariance
a. Definition
b. Example 1: how do you prove
c. Can you give me an example of something that is not time-invariant?
(i) Thermal effects
(ii) Network response time (thinking more generally)
3. Causal systems
a. Definition
b. Example
c. How can something be non-causal? (focus on DSP)
4. Stability
a. Bounded-input bounded-output stability
b. Examples
(i) delay
(ii) squarer
(iii) integrator
C. Linear time-invariant (LTI) System Analysis in the Time Domain
1. Back to "The Box": What if a system is LTI?
2. The impulse response h(t)
a. Defined: response to an impulse
b. The convolution integral
c. LTI system *completely* characterized by h(t)
3. LTI Systems and other properties
a. Causality: condition on h(t)
b. Stability: condition on h(t)
3. Convolution
a. Flip-shift-multiply-add
b. Examples
(i) A lot of pain
(ii) What h(t) do you want to get this y(t) from x(t): ugh...
c. Linearity
D. LTI System Analysis in the Frequency Domain: The Fourier Transform
1. Definition
2. Why does it matter?
a. Convolution -> multiplication
b. Thinking in the frequency domain
c. Examples - you will not do too many of these
(i) 1/T e^{-t/T}
- math
- circuit interpretation and freq domain
3. **Properties**
a. Convolution (of course)
b. Scaling
(i) Property
(ii) p(t) <--> sinc (f)
- math
- bandwidth
- bandwidth versus pulsewidth
(and applications to cell phones)
c. Duality
d. Shifting
(i) Property
(ii) Example - and (no) change in magnitude response
e. Linearity
f. Conjugate symmetry for real signals
g. Modulation (or multiplication)
(i) General property
(ii) Frequency-shifting version
h. Parseval's Theorem
i. Differentiation
4. Example
4. Fourier Series
a. Does the Fourier transform always exist?
a. When to use Fourier Series
b. Definition
5. AM Radio Example
E. Applications: Radio
III. Discrete-Time Signals and Systems
A. Signals
1. Definition of a DT signal: defined at the *integers*
(Section 3.1)
2. Operations (Section 3.2)
a. Time-shift
b. Time reversal
c. Decimation
d. Adding samples
- Expanded (just zeroes in between)
- Interpolated
3. Special Functions (Section 3.3)
a. Discrete delta function: so easy!
b. Discrete unit-step function
c. Sinusoids: A cos(Omega n + theta)
- same sinusoid if you increase Omega by 2 pi
- period often infinity! (sequence does not repeat for Omega rational)
B. Systems
1. Linearity and Time Invariance (Section 3.4.1)
a. Linearity definition
b. Time-invariance definition
c. LTI ==> y[n] = h[n] * x[n] (simple proof in DT!) (Section 3.8)
d. Discrete convolution
2. Causality and stability (Section 3.4.1)
a. Definitions
b. For LTI systems
C. Frequency Analysis
1. Definition of DTFT (Section 9.2)
a. Defined
b. Periodic in 2 pi
c. Y(Omega) = H(Omega) X(Omega)
2. Properties (Section 9.2.1, 9.3)
a. Linearity
b. Conjugate symmetry
c. Time reversal
d. Multiplication by a ramp (n x[n])
e. Time shifting
- property defined
- finding H(Omega) for h[n] a sum of delta functions
f. Freq shifting
- property defined
- x[n]=(-1)^n through filter H(Omega) from previous example
g. Parseval's theorem
- energy and power defined
- theorem proof and statement
3. Important example: sinusoid through H(\Omega)