Overview of Signal and System (by香港中文大学)

Signals and SystemsPart
I. Overview
of Signals
and
Systems
Signal:a function of one or more independent variables; typically contains information about the behavior or nature of some physical phenomena.e.g. V oice (audio), TV (audio + video), light, voltage, current, stock price, etc.• System characterizationhow it responds to input signal(e.g. human auditory system)• System designto process signal in a particular way(e.g. signal restoration, signal identification, image processing)• Audio (intensity vs. time)characteristics: volume, rhythm, pitch• Biomedical signal (voltage vs. time) • e.g. Electrocardiogram• Traffic flow (quantity vs. time)• volume, composition, pattern, mobility;• system involved: traffic lights, roads, junctions • Network Throughput• # of packets/sec., pattern• Stock price (index vs. time; $ vs. time) (Reproduced with permission from Hong Kong Telecom IMS)• Picture (intensity vs. space)• on film -- continuous space, continuous tone• on news paper -- half-tone• on computer -- discrete, quantized• Image (measurement over space)• medical images -- X-ray, CT, ultrasound, MRI • satellite images -- weather map• seismic image -- earthquakey(t)
=
x(1.5t
+
1)
Work
out
a
few
points:
• y(0)
=
x(1)
• y(1)
=
x(2.5) • y(2)
=
x(4)
To
get
from
y
to
x,
we
first
scale
t,
then
shiL.
Therefore,
to
get
from
x
to
y,
we
first
shiL
and
then
scale. • Any signal can be broken into sum of one even and one odd signal.x(t)= Ev {x(t)} + Od {x(t)}Q: How to find the even (odd) part of a signal.A: Ev {x(t)} = (1/2) [x(t) + x(-t)]Od {x(t)} = (1/2) [x(t) - x(-t)]Complex Exponential• For continuous-time complex exponential e jωo t(1) the larger the ωo, the higher the rate of oscillation(2) e jωo t is periodic for any value of ωo.• Are the above two statements still valid for the discrete case e jΩo n ?(1) e j(Ωo+2π)n = e jΩo n e j 2πn = e jΩo n∴ Signal with frequency Ωo= Signal with frequency Ωo+2π, or Ωo +4π, …(Note it is not saying that e jΩo n is periodic with period 2π. Why?)→ To discuss discrete exponential, only need to consider any interval of 2πComplex ExponentialMme
Complex
ExponenMal
(2)
Is
e jΩo n
always
periodic? 
For
a
signal
e jΩo n
to
be
periodic
with
period
N
(>0),
we
must
have
e jΩo(n+N)=
e jΩo n

∴
e jΩo N=1

→
Ωo N
is
mulMple
of
2π
→
Ωo N
=
2π
m where
m
is
an
integer,
or Ωo
/2π =
m/N ⇒
e jΩo N
is
periodic
if
Ωo
/2πis
a
raMonal
number
• Memory/Memoryless• Invertibility and Inverse System • Causality• Stability• Time-invariance• Linearity• A system is causal if the output at anytime only depends on the input at the present time and before.E.g. y[n]=x[n]-x[n-1] : causal y(t)=x(t+1) : non-causal• Causal property is more important for real-time processing. • But for some applications, such as image-processing, no need to process the data causally.E.g.Q:Memoryless ↔ Causal ?• A system is time-invariant if a time shift in the input only causes a time shift in the output.• i.e. If x[n] → y[n], then x[n-n0] → y[n-n0]• Ex.1 y(t)=sin(x(t))Let y1(t)= sin(x1(t)), x2(t)=x1(t-t0)Then y2(t)=sin(x2(t))=sin(x1(t-t0)) = y1(t- t0)∴time-invariant (T.I.)cont.)• Ex.2 y[n]=n x[n]Let y1[n]= n x1[n] & x2[n]=x1[n-n0]Then y2[n]=n x2[n]=n x1[n-n0]However y1[n-n0]=(n-n0) x1[n-n0]≠ y2[n]∴not time-invariant (T.I.)• A system is linear if1. the response to x1(t)+x2(t) is y1(t) +y2(t)--- additivity2. the response to a·x1(t) is a·y1(t), where a is any complex constant.--- scaling• Combine the above two properties, we can concludea·x1(t)+ b·x2(t) ⇒ a·y1(t) + b·y2(t)--- superposition propertyFor discrete-time :a·x1[n]+ b·x2[n] ⇒ a·y1[n] + b·y2[n]cont.)• If linear, zero input gives zero output.(proved by using scaling property)Q: Is y[n]=2x[n]+3 linear? A: No, because it violates zero-in zero-out property. However , this system belongs to the class of “incremental linear system” : difference of output is a linear function of difference of input. y 1[n]-y 2[n]=2x 1[n]+3-{2x 2[n]+3} =2{x 1[n]-x 2[n]}Proof:0=0·x[n]=>0 · y[n]=0• (1). y[n]=x2[n]A: nonlinear; time-invariant• (2). y[n]=n·x[n]A: memoryless; causal; linear; not time-invariant; not stable• Q: Why LTI is so important?• A: So we can represent any input by a summation ofbasic signal (δ(t) or δ[n]) and its time shifted version.Then by using LTI property, the output can be foundfrom the summation of the individual output of eachbasic signal and its times-shifted.。