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2006 Fall
Grading PolicyGrading Policy
Homework – 10%Lab – 20%Midterm Exam – 20%Final Exam – 50%Project
2006 Fall
What will we learn?What will we learn?This course is about using mathematical
techniques to help analyze and synthesize systems which process signals.
We will learn an analytical framework:A language for describing signals and
systems;A set of tools for analyzing signals and
systemsProblems of signal and system analysis:Analyzing existing systems;Designing systems.
2006 Fall
Examples of signals Electrical signals --- voltages and currents in a
circuit Acoustic signals --- audio or speech signals
(analog or digital) Video signals --- intensity variations in an image
(e.g. a CAT scan) Biological signals --- sequence of bases in a gene Economical signals ---price of stocks
…
we will treat noise as unwanted signals
2006 Fall
Examples of systemsExamples of systems
Electrical systems --- amplifer circuitComputer systems --- mp3 playerControl systems --- automobile Economical systems --- stock market
2006 Fall
Signals and SystemsSignals and Systems
Signals : functions of one or more independent variables.
Systems : respond to particular signals by producing other signals or some desired behavior.
As functions, they should have have define domain and range.
2006 Fall
Signal ClassificationTypes of Independent Variable
Time is often the independent variable. Example: the electrical activity of the heart recorded with chest electrodes –– the electrocardiogram (ECG or EKG).
2006 Fall
The variables can also be spatial
Eg. Cervical MRI– In this example, the
signal is the intensity as a function of the spatial variables x and y.
2006 Fall
Independent Variable Dimensionality
An independent variable can be 1-D (t in the EKG) or 2-D (x, y in an image).
We focus on 1-D for mathematical simplicity but the results can be extended to 2-D or even higher dimensions. Also, we will use a generic time t for the independent variable,whether it is time or space.
2006 Fall
Continuous-time (CT) Signals and Discrete-time (DT) Signals
A continuous-time signal will contain a value for all real numbers along the time axis.
X(t)
A discrete-time signal will only have values at equally spaced intervals along the time axis.
X[n] Why DT ? x(t)---sampling--->x[n]
Can be processed by modern digital computers and digital signal processors (DSPs).
2006 Fall
Signal Energy and PowerSignal Energy and Power
Total Energy– Total energy over the time interval
– Total energy over an infinite time interval
2
1
2)(
t
tdttx
2
1
2][
n
n
nx
dttx
2)(
n
nx2
][
Time-Averaged Power （平均功率）– Average power over the time interval– Average power over an infinite time interval
Instantaneous power（瞬时功率）
2006 Fall
Signal Energy and PowerSignal Energy and Power
Energy signal
0 t
tx
0 t
tx
Power signal
0 t
tx
Neither energy,nor power signal
2006 Fall
Right- and Left-Sided Signals
A right-sided signal is zero for t < T and a left-sided signal is zero for t > T, where T can be positive or negative.
2006 Fall
Bounded and Unbounded Signals
Whether the output signal of a system is bounded or unbounded determines the stability of the system.
2006 Fall
Signal OperationSignal OperationShifting y(t)=x(t-t0)
– t0>0 (delay)– t0<0(advance)
Reflecting y(t)=x(-t)Time-Scaling y(t)=x(at)
– a>1(compress) – a<1(expand)
It is rarely use the time-scaling operation when dealing with discrete waveform.
2006 Fall
Signal OperationSignal Operation
)]([)(a
btafbatf
)]([)(a
btafbatf
ab
(1)Contract by a;(2) Shift right with – sign, shift left with + sign.
ab
ab
(1)Reflect and contract by a;(2) Shift right with – sign, shift left with + sign.
ab
2006 Fall
Signals with symmetryPeriodic signals
– CT x(t) = x(t + T)– DT x[n] = x[n + N]
If T1/T2=q/r, where q and r are integers, then there is T=rT1=qT2, which makes
Demo: sum of periodic signals
)()()()( 2121 txtxTtxTtx
2006 Fall
Signals with symmetry (continued)
Even and odd signals– Even x(t) = x(-t) or x[n] = x[-n]– Odd x(t) = -x(-t) or x[n] = - x[-n]
x(0) = 0 or x[0] = 0
Any signals can be expressed as a sum of Even and Odd signals. That is:
.2)]()([)(
,2)]()([)(
)()()(
txtxtx
txtxtx
where
txtxtx
odd
even
oddeven
2006 Fall
Summary What we have learned in this lecture?
– Examples and Classification of Signal– Signal operations – Signals with symmetry
What was the most important point in the lecture? What was the muddiest point? What would you like to hear more about?
2006 Fall
ReadlistReadlist
Signals and Systems – 1.3,1.4,2.5, Mathematical Review (P53)
Question:– Periodic of DT Signals– Euler Relation
2006 Fall
Problem SetProblem Set
1.21(a),(c)1.22(b),(d)1.23(a)1.24(b)
