1. Signal and Systems Prof. H. Sameti
Chapter #1:
Signals
Systems
Some examples of systems
System properties and examples
Causality
Linearity
Time invariance
Reformatted version of open course notes from MIT opencourseware

2. Book Chapter#: Section#
Signals
Signals are functions of independent variables that carry information. For example:
Electrical signals voltages and currents in a circuit
Acoustic signals audio or speech signals(analog or digital)
Video signals intensity variations in an image
Biological signals sequence of bases in a gene
Department of Computer Engineering, Signal and Systems

3. The Independent Variables
Book Chapter#: Section#
The Independent Variables
Can be continuous
Trajectory of a space shuttle
Mass density in a cross-section of a brain
Can be discrete
DNA base sequence
Digital image pixels
Can be 1-D, 2-D, ••• N-D
For this course: Focus on a single (1-D) independent variable which we call "time”
Continuous-Time (CT) signals: x(t), t continuous values
Discrete-Time (DT) signals: x[n] , n integer values only
Computer Engineering Department, Signal and Systems

4. Book Chapter#: Section#
CT Signals
Most of the signals in the physical world are CT signals—E.g. voltage & current, pressure, temperature, velocity, etc.
Computer Engineering Department, Signal and Systems

5. DT Signals x[n], n—integer, time varies discretely
Book Chapter#: Section#
DT Signals
x[n], n—integer, time varies discretely
Examples of DT signals in nature:
DNA base sequence
Population of the nth generation of certain species
Computer Engineering Department, Signal and Systems

6. Many human-made DT Signals
Book Chapter#: Section#
Many human-made DT Signals
Ex.#1 Weekly
Dow-Jones industrial average
Ex.#2 digital image
Why DT? — Can be processed by modern
digital computers and digital signal
processors (DSPs).
Computer Engineering Department, Signal and Systems

7. Book Chapter#: Section#
SYSTEMS
For the most part, our view of systems will be from an input-output perspective:
A system responds to applied input signals, and its response is described in terms of one or more output signals
Computer Engineering Department, Signal and Systems

8. EXAMPLES OF SYSTEMS An RLC circuit
Book Chapter#: Section#
EXAMPLES OF SYSTEMS
An RLC circuit
Dynamics of an aircraft or space vehicle
An algorithm for analyzing financial and economic factors to predict bond prices
An algorithm for post-flight analysis of a space launch
An edge detection algorithm for medical images
Computer Engineering Department, Signal and Systems

9. SYSTEM INTERCONNECTIOINS
Book Chapter#: Section#
SYSTEM INTERCONNECTIOINS
An important concept is that of interconnecting systems
To build more complex systems by interconnecting simpler subsystems
To modify response of a system
Cascade
Parallel
Feedback
Computer Engineering Department, Signal and Systems

10. SYSTEM EXAMPLES Example. #1 RLC circuit Book Chapter#: Section#
Computer Engineering Department, Signal and Systems

11. Example. #2 Mechanical system
Force Balance:
Observation: Very different physical systems may be modeled mathematically in very similar ways.

12. Example. #3 Thermal system Cooling Fin in Steady State
t = distance along rod
y(t) = Fin temperature as function of position
x(t) = Surrounding temperature along the fin

13. Example. #3 Thermal system (Continued)
Observations
Independent variable can be something other than time, such as space.
Such systems may, more naturally, have boundary conditions, rather than “initial” conditions.

14. Example. #4 Financial system
Fluctuations in the price of zero-coupon bonds
t = 0 Time of purchase at price y0
t = T Time of maturity at value yt
y(t) = Values of bond at time t
x(t)= Influence of external factors on fluctuations in bond price
Observation: Even if the independent variable is time, there are interesting and important systems which have boundary conditions.

15. A Rudimentary “Edge” Detector
Example. #5
A Rudimentary “Edge” Detector
y[n]=x[n+1]-2x[n]+x[n-1]
={x[n+1]-x[n]}-{x[n]-x[n-1]}
= “Second difference”
This system detects changes in signal slope
x[n]=n → y[n]=0
x[n]=nu[n] → y[n]

16. Book Chapter#: Section#
Observations
A very rich class of systems (but by no means all systems of interest to us) are described by differential and difference equations.
Such an equation, by itself, does not completely describe the input-output behavior of a system: we need auxiliary conditions (initial conditions, boundary conditions).
In some cases the system of interest has time as the natural independent variable and is causal. However, that is not always the case.
Very different physical systems may have very similar mathematical descriptions.
Computer Engineering Department, Signal and Systems

17. SYSTEM PROPERTIES (Causality, Linearity, Time-invariance, etc.)
Book Chapter#: Section#
SYSTEM PROPERTIES (Causality, Linearity, Time-invariance, etc.)
WHY?
Important practical/physical implications
They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.
Computer Engineering Department, Signal and Systems

18. Book Chapter#: Section#
CAUSALITY
A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time.
All real-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a noncausal system whose output depends on tomorrow’s stock price.)
Causality does not apply to spatially varying signals. (We can move both left and right, up and down.)
Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast.
Computer Engineering Department, Signal and Systems

19. CAUSALITY (continued)
Book Chapter#: Section#
CAUSALITY (continued)
Mathematically (in CT):
A system x(t) →y(t) is causal if
when x1(t) →y1(t) x2(t) →y2(t)
and x1(t) = x2(t) for all t≤ to
Then y1(t) = y2(t) for all t≤ to
Computer Engineering Department, Signal and Systems

20. CAUSAL OR NONCAUSAL 𝑦(𝑡)= 𝑥 2 (𝑡−1 𝐸.𝑔. 𝑦(5 depends on x(4) … causal
Book Chapter#: Section#
CAUSAL OR NONCAUSAL
𝑦(𝑡)= 𝑥 2 (𝑡−1 𝐸.𝑔. 𝑦(5 depends on x(4) … causal
𝑦(𝑡)=𝑥(𝑡+1 𝐸.𝑔. 𝑦(5)=𝑥(6), y depends on future noncausal
𝑦[𝑛]=𝑥[−𝑛 𝐸.𝑔. 𝑦[5]=𝑥[−5 ok, but
𝑦[−5]=𝑥[5 y depends on future noncausal
𝑦[𝑛]= 𝑛+1 𝑥 3 [𝑛−1] depends on x(4) … causal
Computer Engineering Department, Signal and Systems

21. Book Chapter#: Section#
TIME-INVARIANCE (TI)
Informally, a system is time-invariant (TI) if its behavior does not depend on what time it is.
Mathematically (in DT): A system x[n] → y[n] is TI if for any input x[n] and any time shift n0,
If x[n] →y[n]
then x[n -n0] →y[n -n0]
Similarly for a CT time-invariant system,
If x(t) →y(t)
then x(t -to) →y(t -to) .
Computer Engineering Department, Signal and Systems

22. TIME-INVARIANT OR TIME-VARYING?
Book Chapter#: Section#
TIME-INVARIANT OR TIME-VARYING?
𝑦(𝑡)= 𝑥 2 (𝑡+1 is TI
𝑦[𝑛]= 𝑛+1 𝑥 3 [𝑛−1] is TV (NOT time-invariant)
Computer Engineering Department, Signal and Systems

23. NOW WE CAN DEDUCE SOMETHING!
Book Chapter#: Section#
NOW WE CAN DEDUCE SOMETHING!
Fact: If the input to a TI System is periodic, then the output is periodic with the same period.
“Proof”: Suppose x(t + T) = x(t)
and x(t) → y(t)
Then by TI
x(t + T) →y(t + T).
↑ ↑
So these must be the same output, i.e., y(t) = y(t + T).
These are the same input!
Computer Engineering Department, Signal and Systems

24. LINEAR AND NONLINEAR SYSTEMS
Book Chapter#: Section#
LINEAR AND NONLINEAR SYSTEMS
Many systems are nonlinear. For example: many circuit elements (e.g., diodes), dynamics of aircraft, econometric models,…
However, in this course we focus exclusively on linear systems.
Why?
Linear models represent accurate representations of
behavior of many systems (e.g., linear resistors,
capacitors, other examples given previously,…)
Can often linearize models to examine “small signal”
perturbations around “operating points”
Linear systems are analytically tractable, providing basis
for important tools and considerable insight
Computer Engineering Department, Signal and Systems

25. Book Chapter#: Section#
LINEARITY
A (CT) system is linear if it has the superposition property:
If x1(t) →y1(t) and x2(t) →y2(t)
then ax1(t) + bx2(t) → ay1(t) + by2(t)
y[n] = x2[n] Nonlinear, TI, Causal
y(t) = x(2t) Linear, not TI, Noncausal
Can you find systems with other combinations ?
-e.g. Linear, TI, Noncausal
Linear, not TI, Causal
Computer Engineering Department, Signal and Systems

26. EXAMPLES OF LINEAR SYSTEMS
Book Chapter#: Section#
EXAMPLES OF LINEAR SYSTEMS
Ex. 1:
𝑦(𝑡)= 𝑥 ∗ (𝑡)
Additive, but not scalable, so not linear
Ex. 2:
𝑦 𝑡 = 𝑥 2 (𝑡) 𝑥(𝑡−1)
Scalable, but not additive, so not linear
Computer Engineering Department, Signal and Systems

27. PROPERTIES OF LINEAR SYSTEMS
Book Chapter#: Section#
PROPERTIES OF LINEAR SYSTEMS
Superposition
𝑥 𝑘 [𝑛]→ 𝑦 𝑘 [𝑛 𝑘 𝑎 𝑘 𝑥 𝑘 [𝑛]→ 𝑘 𝑎 𝑘 𝑦 𝑘 [𝑛
For linear systems, zero input → zero output
"Proof" =0⋅x[n]→0⋅y[n]=0 If
Then
Computer Engineering Department, Signal and Systems

28. Properties of Linear Systems (Continued)
Book Chapter#: Section#
Properties of Linear Systems (Continued)
A linear system is causal if and only if it satisfies the condition of initial rest:
𝑥(𝑡)=0 𝑓𝑜𝑟 𝑡≤0→𝑦(𝑡)=0 𝑓𝑜𝑟 𝑡≤ 𝑡 0 (∗
“Proof”
a) Suppose system is causal. Show that (*) holds.
b) Suppose (*) holds. Show that the system is causal.
Computer Engineering Department, Signal and Systems

29. LINEAR TIME-INVARIANT (LTI) SYSTEMS
Book Chapter#: Section#
LINEAR TIME-INVARIANT (LTI) SYSTEMS
Focus of most of this course
- Practical importance (Eg. #1-3 earlier this
lecture are all LTI systems.)
- The powerful analysis tools associated with LTI
systems
A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs
Computer Engineering Department, Signal and Systems

30. Example: DT LTI System Book Chapter#: Section#
Computer Engineering Department, Signal and Systems