Basic concepts in DT systems - UVicaalbu/elec310_2009/ELEC 310-4-Basic...Alexandra Branzan Albu ELEC...

Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1

Basic concepts in DT systems


Readings and homework

For DT systems:• Textbook: sections 1.5, 1.6

• Suggested homework:– pp. 57-58:

• 1.15• 1.16• 1.18• 1.19


Course outline

DT signals (cont’d)– a brief note on combining time-shift and time reversal – Elementary DT signals

• Periodic signals (cont’d)• Unit step function• Unit impulse function

DT systems:– System properties:

• Causality• Stability• Linearity• Invariance


On combining time-shift and time reversal

][][][

][][][

integer positive ][][

) (

) (

αα

α

αα

α

α

+−⎯⎯ →⎯+⎯⎯⎯⎯ →⎯

+−⎯⎯⎯⎯ →⎯−⎯⎯ →⎯

−=

nxnxnxor

nxnxnxle ways:Two possib

nxny

reversalbyleftshift

advance

byrightshiftdelayreversal


Periodicity properties of DT

• 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

The DT signal x[n] is periodic only if ω0/2π is a rational number.

Fundamental frequency where m and N have no factors in common.

Read Table 1.1 for a comparative summary of periodicity properties of CT and DT signals

Nmπω 20 =


Example

• Determine the fundamental period of

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

62cos2

8sin

4cos2][ ππππ nnnnx


Non-periodic elementary signals

• Building blocks for constructing and representing signals and systems

• The unit impulse (unit sample):

• The unit step:

[ ]⎩⎨⎧

=≠

=0n if 10n if 0

nδ

[ ]⎩⎨⎧

≥<

=0n if 10n if 0

nu


Relationships between DT unit impulse and DT unit step

• The DT unit impulse is the first difference of the DT unit-step

• Conversely, the DT unit step is the running sum of the unit sample

]1[][][ −−= nununδ

[ ]

[ ]∑

∑

∞

=

−∞=

−=

=

0][

][

k

n

m

knnu

mnu

δ

δ


Sampling property of the unit impulse sequence

][]0[][][ nxnnx δδ =

];[][][][ 000 nnnxnnnx −=− δδ


CT unit impulse and unit step functions

• CT unit step function:

• CT unit step function is the running integral of the unit impulse:

• Conversely, the unit impulse can be thought as the first derivative of the unit step


CT unit step and unit impulse functions (cont’d)

• u(t) is discontinous at t=0 and formally not differentiable;

• We can imagine a limit process

• with DT signals we do not need continuity, derivability etc.

)()()(0

tdt

tdut δδ ⎯⎯ →⎯=→Δ

ΔΔ


Course outline

DT signals (cont’d)– a brief note on combining time-shift and time reversal – Elementary DT signals

• Periodic signals (cont’d)• Unit step function• Unit impulse function

DT systems:– example – System properties:

• Causality• Stability• Linearity• Invariance


What is a system?


Example of DT systemA rudimentary ‘edge’ detector


Observations

• A very rich class of systems (but by no means all systems of interest to us) are described by differential and difference equations

• Such an equation, by itself, does not completely describe the input-output behaviour of the system: we need auxiliary conditions (initial conditions)

• In some cases the system of interest has time as the natural independent variable and is causal. However, that is not always the case.

• Very different physical systems may have very similar mathematical descriptions.


System properties

• invertibility, causality, linearity, stability, time-invariance etc.

• Why?– Important practical/physical implications– They provide us with structure that we can exploit

for both system analysis and system design


Invertibility

• In an invertible system, distinct inputs lead to distinct outputs

• The original system cascaded with its inverse yield an output w[n] equal to the input x[n] of the original system.

• example


Invertibility analysis

• Determine if the following systems are invertible or not. If yes, construct their corresponding inverse systems

• y[n]=nx[n]

• y[n]=x[1-n]


Causality

• A system is causal if the output does not anticipate future values of the input, i.e. if the output at any time depends only of the values of the input up to that time– All real-time physical systems are causal. Time only

moves forward, and effect occurs after the cause.– Causality does not apply to systems processing

spatially varying signals (we can move both left and right, up and down)

– Causality does not apply to systems processing recorded signals (e.g. taped sports games versus live broadcast)


Causality (cont’d)

• Mathematically: A system x[n] → y[n] is causal if

• Causal or non-causal?– y[n]=x[-n]– y[n]=(1/2)n+1 x3[n-1]

021

021

2211

][][ ][][

[n]][ [n]][

nnfor allnynyThennnfor allnxnxand

ynxandynxwhen

≤=≤=

→→


Stability

• a stable system is one in which bounded inputs lead to responses that do not diverge

• Ex: model for the balance of a bank account from month to month y[n]=1.01y[n-1]+x[n], x[n]>0 for all n>=0.


Stable or unstable?

• y[n]=nx[n]• y[n]=x[4n+1]

• Strategy: • to prove that a system is unstable, we need to

find a specific bounded input that leads to an unbounded output.

• If such an example does not exist, then the system is stable


Time invariance

• Conceptually: A system is time invariant (TI) if its behaviour does not depend on a particular moment in time.

• Mathematically:– for a DT time-invariant system:


Time invariant or time-varying?

y[n]=nx[n]

y[n]=sin(x[n])

Strategy: we must determine whether the time invariance property holds for any input and any time shift

When a system is suspected of being time-varying, we can seek a counterexample (a specific input signal for which the condition of time-invariance would be violated)


Important

• If the input to a TI system is periodic, then the output is periodic with the same period.


Linear and non-linear systems

• Many systems are non-linear (ex: circuit elements: diodes, dynamics of aircraft etc.)

• In ELEC 310 we focus exclusively on linearsystems

• Why?– We can often linearize models to examine ‘small

signal’ perturbations around ‘operating’ points– Linear systems are analytically tractable, providing

basis for important tools used in DSP.


Linearity

• A system is called linear if it has two mathematical properties:

• Let us consider x1[n]→ y1[n] and x2[n]→ y2[n]

Additivity: x1[n] + x2[n] → y1[n] + y2[n]Homogeneity (scaling): ax1[n] → a x1[n]

We can combine these two properties into one:ax1[n]+ bx2[n] → ay1[n] + by2[n]


Properties of linear systems

• Superposition

• For linear systems, zero input → zero output• A linear system is causal if and only if it satisfies

the condition of initial rest

0 0][0 0][ ≤=→≤= tfornytfornx


Linear Time-Invariant (LTI) Systems

• Our focus for most of this course• A basic fact: If we know the response of an

LTI system to some inputs, we actually know its response to many inputs


System interconnections

• Serial• Parallel• Feedback• example


You know now

• the primary focus in this class is on linear, time-invariant LTI systems in the DT domain

• LTI systems are defined in a similar way in both CT and DT domains

• How to compute the global input-output function of interconnected systems

• How to determine whether a system is:– Causal– Stable– Time-invariant– Linear– Invertible

Basic concepts in DT systems - UVicaalbu/elec310_2009/ELEC 310-4-Basic...Alexandra Branzan Albu ELEC...

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