Leo Lam © 2010-2012

Signals and SystemsEE235

Leo Lam © 2010-2012

An ex and a Constant were…

…walking down the street; when a Differentiator walked up to them.

Constant started running away, and ex asked him, “what are you doing?!”

Constant replied, “If I meet a Differentiator, I will disappear!”

ex said, proudly, “I don’t care, I am ex!”, and walked up to the Differentiator.

“Hi I am ex,” he said, thumbing his nose…“Hi,” said the Differentiator, “I’m d/dy.”

Leo Lam © 2010-2012

Today’s menu

• Textbook Chapter 1, Schaum’s Chapter 1• To do:

– Sign up to Facebook Group– Bookmark our website

• From yesterday: definitions• End of hand-waving• Describing Common Signals

– Type of signals– Some standard signals

• Periodicity

Leo Lam © 2010-2012

Signals:

• A signal is a mathematical function– x(t)– x is the value (real, complex) y-axis– t is the independent variable (1D, 2D etc.) x-axis– Both can be Continuous or Discrete– Examples of x…

Leo Lam © 2010-2012

Signal types

• Continuous time / Discrete time– An x-axis relationship

• Discrete time = “indexed” time

Leo Lam © 2010-2012

Signals: Notations

• A continuous time signal is specified at all values of time, when time is a real number.

( ),x t t R

Leo Lam © 2010-2012

Signals: Notations

• A discrete time signal is specified at only discrete values of time (e.g. only on integers)

[ ],x n nZ

Leo Lam © 2010-2012

What types are these?

1) 90.3 FM radio transmitted signal2) Daily count of orcas in Puget Sound3) Muscle contraction of your heart over time 4) A capacitor’s charge over time5) A picture taken by a digital camera6) Local news broadcast to your old TV7) Video on YouTube8) Your voice

(c)

((c))

(c)

(continuous)

(c)(d)

(d)

(discrete)

Leo Lam © 2010-2012

Analog / Digital values (y-axis)

• An analog signal has amplitude that can take any value in a continuous interval (all Real numbers)

[ ] ,x n R n Z

( ) ,x t R t R Where Z is a finite set of values

Leo Lam © 2010-2012

Analog / Digital values (y-axis)

• An digital signal has amplitude that can only take on only a discrete set of values (any arbitrary set).

Where Z and G are finite sets of values

( ) ,x t G t R

[ ] ,x n G n Z

Leo Lam © 2010-2012

Nature vs. Artificial

• Natural signals mostly analog• Computers/gadgets usually digital (today)• Signal can be continuous in time but discrete

in value (a continuous time, digital signal)

Leo Lam © 2010-2012

Brake!

• X-axis: continuous and discrete• Y-axis: continuous (analog) and discrete

(digital)• Our class: (mostly) Continuous time, analog

values (real and complex)• Clear so far?

Leo Lam © 2010-2012

Common signals (memorize)

• Building blocks to bigger things

constant signal

t

a ( )x t a

0

unit step signal

t

1

0

unit ramp signal

t

1

u(t)=0 for t<0u(t)=1 for t≥0

r(t)=0 for t<0r(t)=t for t≥0r(t)=t*u(t) for t≥0

t

dutr )()(

Leo Lam © 2010-2012

Sinusoids/Decaying sinusoids

Leo Lam © 2010-2012

Decaying and growing

Leo Lam © 2010-2012

Generalizing the sinusoids

General form: x(t)=Ceat, a=σ+jω

Equivalently: x(t)=Ceσtejωt

Remember Euler’s Formula?

x(t)=Ceσtejωt

)sin()cos( tjte tj

amplitude

Exponential(3 types)

Sinusoidal with frequency ω (in radians)

What is the frequency in Hz?

Leo Lam © 2010-2012

Imaginary signals

zr

a

b

z=a+jb real/imaginaryz=rejΦ magnitude/phase

f

real

imag

Remember how to convert between the two?