basic concepts in dt signals - electrical engineeringaalbu/elec310_2009/elec310-2-3 basic...

Alexandra Branzan Albu ELEC 310-Spring 2009-Lectures 2 and 3 1

Basic concepts in DT signals


Textbook readings

• Textbook: sections 1.1, 1.2, 1.3, 1.4

• Suggested drill exercises (not marked):– pp. 57-58:

• 1.1, 1.2• 1.3 d) e) f) • 1.4• 1.6 b) c)• 1.7• 1.9 c) d) e)• 1.11


Course outline

• Signals: mathematical description and representation– Energy and power (1.1.2)– Transformations of a signal– Periodic signals (1.2.2)– Even and odd signals (1.2.3)– Exponential and sinusoidal signals (1.3)

• CT and DT– Basic DT signals (1.4.1)

• The DT unit impulse and the unit step function– Basic CT signals (1.4.2)

• The CT unit impulse and the unit step function


Math prerequisites

• Make sure that you are familiar with complex numbers, their representation in polar and Cartesian coordinates, Euler’s relation etc.– Page 71-72 textbook– read first-aid kits posted on the web site; work

their examples– p. 57 basic problems with answers: 1.1, 1.2


What are signals?

• functions of one or more independent variables

• This course: one-dimensional discrete-time (DT) signals

• We will generally refer to the independent variable as time


Signal energy and power• Notions that were initially applicable to signals produced by

physical systems only (i.e. v(t), i(t) across a resistor of resistance R)

• Instantaneous electric power

• Energy expended over [t1, t2]

• Generalization: The energy and power characterize any type of signals (not only electrical)

• The energy of a CT signal x(t) over [t1, t2] • average power

• The energy of a DT signal x[n] over [n1, n2] • average power


Signal energy and power (cont’d)

• Energy and power over an infinite time interval• Total energy for a CT signal:

• Total energy for a DT signal:

• Time-averaged power over an infinite time interval for CT and DT


Classes of signals (energy viewpoint)

• In terms of total energy and average power, we can identify three important classes of signals:

• 1) Finite total energyE∞< ∞ →

• 2) Finite average power0<P∞< ∞ → E∞= ∞

• 3) Infinite P∞, E∞

02

lim == ∞∞→

∞ TEP

T


Examples

• Decide whether x1 [n], x2 [n] given below are energy signals or power signals

( ) 0 5.03][1 ≥= nnx n

nnx allfor 4][2 =


Transformations of a signal

• Operations on the independent variable: – time shift, – time scale, – time reversal

• Operations on the values of the signal: – scaling (amplification), – discretization– differentiation etc.

Slide adapted from E. Cretu, Lecture notes on Signals and Systems, UBC 2007.


Time-shifting

a) CT signal : rectangular pulse of amplitude 1.0 and duration 1.0

b) time-shifted (delayed) version of a) by t0=2.0

x(t)→x(t-t0)x[n]→x[n-n0]n0>0 → x[n-n0] is the delayed version of x[n]n0<0 → x[n-n0] is the advanced version of x[n]

Time shifting does not change the shape of the signalExample: delay in the propagation of a signal through a material medium


Periodic signals

Periodic signals are invariant to certain time shifts.CT case:

– x(t)=x(t+T) for all values of t, T>0.– fundamental period: T0

DT case– x[n]=x[n+N] for all values of n, N a positive integer– fundamental period: N0

• What is the total energy of a periodic signal?• How can we determine if a signal is periodic or not?

– Direct approach: try to find N so that x[n]=x[n+N] for all N– Decompose the signal into a sum of periodic signals (sinusoids

or cosinusoids)


Transformations of a signal: Time reversal

• Reflection about the vertical axis (t=0;n=0)– y(t)=x(-t)– y[n]=x[-n]

• See figure 1.10 p. 9 for DT time reversal• Useful for:

– Classification of signals from the time-reversal viewpoint– Performing DT (and CT) convolution


Even and odd signals

• Symmetry under time-reversal• CT even signal x(-t)=x(t)• DT even signal x[-n]=x[n]

• CT odd signal x(-t)=-x(t)• DT odd signal x[-n]=-x[n]

• Any signal can be broken down into the sum of an even and an odd signal

2)()()(;

2)()()(

)()()(

txtxtxtxtxtx

txtxtx

oddeven

oddeven

−−=

−+=

+=ΔΔ


Decomposition into odd-even parts

from http://cnx.org/content/m10057/latest/


Sum of the odd-even parts

from http://cnx.org/content/m10057/latest/


Time scaling (CT signals)

• x(t)→y(t)=x(at)• a<1: linear stretching• a>1: linear compression • example: audio tape played at double/half speed• Time scaling modifies the frequency spectrum


Time scaling (DT signals)y[n]=x[an], a>0

If k>1 (compression) the transformation is irreversible- some values of the originals signal are lost

Slide adapted from E. Cretu, Lecture notes on Signals and Systems, UBC 2007.


Example

• 1.22 b), c) and d)


Importance of transformations of independent variable

• Naturally occur in physical systems• Are useful for identifying specific categories

of signals (and systems)• Will be further used in specifying important

properties of the Fourier transform and Z transform


Elementary signals

• Are basic building blocks for synthesizing/decomposing a wide range of more complex signals

• What types of signals should be chosen as primitives?– It depends on the class of signals/systems one wants to

analyze/design/simulate– ELEC 310 deals with Linear Time Invariant Systems – For this class of systems exponential and sinusoidal

functions are shape-invariant– It is useful to consider these functions as elementary

building blocks


Exponential and sinusoidal signals• Complex exponential signal x(t)=Ceat, C and a complex• Some cases of particular interest:

– C real, a real: real exponentials– a=+jω0 or a=-j ω0 x(t) is periodic – a complex, C complex: general complex exponential signals

• Periodic exponentials will serve for building representations of other signals

• We will use sets of harmonics, all of which are periodic with a common period T0

( ) ,...2,1,0,0 ±±==Φ ket tikk

ω


CT real exponential signals

RaCCetx at ∈= ,,)(a<0 decaying exponential a>0 growing exponential (step response of an RC circuit) (chain reaction)


DT real exponential signals

nan CCenx α==][0<α<1 Decaying exponential

α>1 Growing exponential


DT real exponential signals (cont’d)

-1<α<0

α<-1

nan CCenx α==][


Periodic signals: exponential and sinusoidal forms

)cos(][ 0 φω += nAnxGiven

we want to express it as a sum of complex exponentials


General complex exponential signals

0ω

θ

αα j

j

e

eCC

=

=


Periodicity properties of DT

• they are different from CT

][ )( 00 njtj enxetx ωω ==

1) The larger the magnitude of ω0, the higher is the rate of oscillation in the signal

2) This signal is periodic for any non-zero value of ω0


Periodicity properties of DT

][ )( 00 njtj enxetx ωω ==1) The larger the magnitude of

ω0, the higher is the rate of oscillation in the signal

( ) njnj ee 00 2 ωπω =+

The signal with frequency ω0 is identical to signals with frequencies ω0 ± 2kπ , k integerWe need to choose an interval of length 2π , either [- π, π) or [0,2 π).

How does the rate of oscillation vary from 0 to 2π?

basic concepts in dt signals - electrical engineeringaalbu/elec310_2009/elec310-2-3 basic...

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