EE-2027 SaS, L2 1/25

Lecture 2: Signals Concepts & Properties

(1) Systems, signals, mathematical models. Continuous-time and discrete-time signals. Energy and power signals. Causal, Linear and Time Invariant systems.

Specific objectives for this lecture include• Energy and power for continuous & discrete-time

signals

EE-2027 SaS, L2 2/25

Lecture 2: Resources

• SaS, O&W, Sections 1.1-1.4• SaS, H&vV, Sections 1.4-1.9

EE-2027 SaS, L2 3/25

Reminder: Continuous & Discrete Signals

x(t)

t

x[n]

n

Continuous-Time SignalsMost signals in the real world are

continuous time, as the scale is infinitesimally fine.

E.g. voltage, velocity, Denote by x(t), where the time interval

may be bounded (finite) or infiniteDiscrete-Time SignalsSome real world and many digital

signals are discrete time, as they are sampled

E.g. pixels, daily stock price (anything that a digital computer processes)

Denote by x[n], where n is an integer value that varies discretely

Sampled continuous signalx[n] =x(nk)

EE-2027 SaS, L2 4/25

“Electrical” Signal Energy & Power

It is often useful to characterise signals by measures such as energy and power

For example, the instantaneous power of a resistor is:

and the total energy expanded over the interval [t1, t2] is:

and the average energy is:

How are these concepts defined for any continuous or discrete time signal?

)(1

)()()( 2 tvR

titvtp

2

1

2

1

)(1

)( 2t

t

t

tdttv

Rdttp

2

1

2

1

)(11

)(1 2

1212

t

t

t

tdttv

Rttdttp

tt

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Generic Signal Energy and Power

Total energy of a continuous signal x(t) over [t1, t2] is:

where |.| denote the magnitude of the (complex) number.

Similarly for a discrete time signal x[n] over [n1, n2]:

By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the average power, P

2

1

2)(

t

tdttxE

2

1

2][

n

nnnxE

EE-2027 SaS, L2 6/25

Energy and Power over Infinite TimeFor many signals, we’re interested in examining the power and energy

over an infinite time interval (-∞, ∞). These quantities are therefore defined by:

If the sums or integrals do not converge, the energy of such a signal is infinite

Two important (sub)classes of signals

1. Finite total energy (and therefore zero average power)

2. Finite average power (and therefore infinite total energy)

Signal analysis over infinite time, all depends on the “tails” (limiting behaviour)

dttxdttxET

TT

22)()(lim

n

N

NnN nxnxE22][][lim

T

TT dttxT

P2)(

2

1lim

N

NnN nxN

P2][

12

1lim

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Exponential & Sinusoidal Signal PropertiesPeriodic signals, in particular complex periodic

and sinusoidal signals, have infinite total energy but finite average power.

Consider energy over one period:

Therefore:

Average power:

Useful to consider harmonic signals

Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency

00

0

2

0

001

T j tperiod

T

E e dt

dt T

11

0

periodperiod ET

P

E

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General Complex Exponential Signals

So far, considered the real and periodic complex exponential

Now consider when C can be complex. Let us express C is polar form and a in rectangular form:

So

Using Euler’s relation

These are damped sinusoids

0

jra

eCC j

tjrttjrjat eeCeeCCe )()( 00

))sin(())cos(( 00)( 0 teCjteCeeCCe rtrttjrjat

EE-2027 SaS, L2 9/25

Discrete Unit Impulse and Step Signals

The discrete unit impulse signal is defined:

Useful as a basis for analyzing other signals

The discrete unit step signal is defined:

Note that the unit impulse is the first difference (derivative) of the step signal

Similarly, the unit step is the running sum (integral) of the unit impulse.

01

00][][

n

nnnx

01

00][][

n

nnunx

]1[][][ nunun

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Continuous Unit Impulse and Step Signals

The continuous unit impulse signal is defined:

Note that it is discontinuous at t=0

The arrow is used to denote area, rather than actual value

Again, useful for an infinite basis

The continuous unit step signal is defined:

0

00)()(

t

tttx

tdtutx )()()(

01

00)()(

t

ttutx

EE-2027 SaS, L2 11/25

Written Assignment 1

Q1.3

Q1.4

Q1.5

Q1.6

Q1.13