outline introduction to discrete time signals and systems

Introduction toDiscrete Time Signals and Systems

Week 02, INF3190/4190

Andreas Austeng

Department of Informatics, University of Oslo

August 2019

AA, IN3190/4190 (Ifi/UiO) Signals and Systems Aug. 2019 1 / 46

Outline

Outline

1 Discrete-time signalsTime-domain representationSome elementary discrete-time signalsClassification of sequencesSignal manipulations

2 Discrete-time systemsDiscrete-time systemsSystem propertiesFIR and IIR LTI SystemsDiscrete-time systems described by difference equations


Discrete-time signals

Outline




Discrete-time signals Time-domain representation

Discrete-time sequences

Signals represented as sequence of numbers / samples:{x [n]} = {. . . , x [�1], x

"[0], x [1], . . .}.

x [n] represents the n’th sample of sequence {x [n]}N2N1

.When clear, we let x [n] represent either the n’th sample or entiresequence.x [n] only defined for integer values of n, and undefined fornon-integer values of n.



Discrete-time systemsProcesses a given input sequence x [n] to generate anoutput sequence y [n].In most cases; a single-input, single output system

I Characterized through an input-output transformation:

y [n] = H{x [n]}

or x [n] �! H{·} �! y [n]

or x [n] H�! y [n].

Examples: Constant multiplier, unit delay, unit advance.I Visualized with a block diagram

I Example 2-input, 1-output systems: Adder, Signal multiplier(Modulator).



Sampled sequences

x [n] = {. . . , x [�1], x"[0], x [1], . . .}.

x [n] may be generated by periodically samling a continuous-timesignal xa(t) at uniform intervals of time, i.e.x [n] = xa(t)|t=nT = xa(nT ), n = . . . ,�3,�2,�1, 0, 1, 2, . . .T : sampling interval/period, FT = 1

T : sampling frequency,⌦T = 2⇡FT : Sampling angular frequency.



Sampled sequences ...



Sampled sequences ...



Sampled sequences

Example (Sampling of continuous-time signal)Consider the continuous-time signalx(t) = A cos(2⇡f0t + �) = A cos(⌦0t + �)

The corresponding discrete-time signal is

x [n] = A cos[⌦0nT + �]

= A cos[2⇡⌦0⌦T

n + �]

= A cos[!0n + �],

where !0 = 2⇡⌦0/⌦T = ⌦0T is the normalized angular frequencyof x [n].



Example (continues ...)Given three signals g1(t) = cos(6⇡t), g2(t) = cos(14⇡t) andg3(t) = cos(26⇡t) with frequencies 3 Hz, 7 Hz and 13 Hz.When sampled at a samplings rate of 10 Hz, we get the threesequences g1[n] = cos[0.6⇡n], g2[t ] = cos[1.4⇡n] andg3[n] = cos[2.6⇡n].

Note: The same sample values for any given n.



Example (continues ...)Another example of aliasing ...



Example (continues ...)

Theorem (Shannon Sampling Theorem)A continuous-time signal x(t) with frequencies no higher than fmax canbe reconstructed exactly from its samples x [nT ] = x(nTs), if thesamples are taken at a rage fs = 1/Ts that is greater than 2fmax .

2fmax : The Nyquist rate.


Discrete-time signals Some elementary discrete-time signals

General discrete-time signals

x [n] = {. . . , x [�1], x"[0], x [1], . . .}

I May have infinite length.I Infinite-length sequences may further be classified as either being

right-sided, left-sided, or two-sided.A finite-length sequence is equal to zero for all values of n outsidethe interval [n1, n2].

I Length of such a sequence: Lx = n2 � n1 + 1.

In general, a discrete-time signal may be complex-valued.



Complex sequences

A complex signal may be expressedeither in terms of its real and imaginary parts,

z[n] = a[n] + |b[n] = <{z[n]}+ |={z[n]}or in polar forms in terms of its magnitude and phase,

z[n] = |z[n]|exp{|arg{z[n]}}.The magnitude square may be derived as

|z[n]|2 = z[n]z⇤[n] = <2{z[n]}+ =

2{z[n]}.

The phase may be found usingarg{z[n]} = tan�1 ={z[n]}

<{z[n]} .

z⇤[n] is called the complex conjugate, and formed by changing thesign of the imaginary part of z[n]:

z⇤[n] = <{z[n]}� |={z[n]} = |z[n]|exp{�|arg{z[n]}}.



Some elementary discrete-time signals

Unit sample sequence, �[n]. Details

I Any arbitrary sequence x [n] can be synthesized as a weighted sumof delayed and scaled unit-sample sequences.

Unit step sequence, u[n]. Details

Unit ramp sequence, ur [n]. Details

Exponential sequences, x [n] = A↵n8n.

I Real-valued exponential sequences. Details

I Complex-valued exponential sequences. Details

Sinusoidal sequences, x [n] = Acos[!0n + �], 8n. Details

Periodic sequences. Details



The unit sample sequence

�[n] =

(1, n = 00, otherwise,

or �[n] = u[n]� u[n � 1]

−4 −3 −2 −1 0 1 2 3 4 5−0.5

00.5

11.5

Plot of δ[n]

−4 −3 −2 −1 0 1 2 3 4 5−0.5

00.5

11.5

Plot of δ[n−2]

n

Return



The unit step sequence

u[n] =

(1, n � 00, otherwise,

or u[n] =P1

k=0 �[k ]

−4 −3 −2 −1 0 1 2 3 4 5−0.5

00.5

11.5

Plot of u[n]

n

Return



The unit ramp sequence

ur [n] =

(n, n � 00, otherwise.

−4 −3 −2 −1 0 1 2 3 4 50246

Plot of u[n]

n

Return



Real valued exponential sequencesx [n] = A↵n, �1 < n < 1,where A and ↵ are real numbers.A one-sided exponential sequence ↵n, n � 0; ↵ 2 < is called ageometric series.

Geometric sequence:PN

n=0 ↵n�!

1�↵N

1�↵ , 8↵

0 10 20 30 400

10

20

30

40

50α = 1.1

Time index n

Am

plitu

de

0 10 20 30 400

0.2

0.4

0.6

0.8

1α = 0.9

Time index n

Am

plitu

de

Return



Complex valued exponential sequencesx [n] = A↵n, �1 < n < 1,where one or both of A and ↵ are complex numbers.Rewriteing: ↵ = e(⇢0+|!0) and A = |A|e|�

=) x [n] = |A|e|�e(⇢0+|!0)n = x<[n] + |x=[n], wherex<[n] = |A|e⇢0n cos[!0n + �]

x=[n] = |A|e⇢0n sin[!0n + �]

0 10 20 30 40−1

−0.5

0

0.5

1Plot of ’real(exp(−1/10 + j*π/5)*n)’

Time index n

Am

plitu

de

0 10 20 30 40−1

−0.5

0

0.5

1Plot of ’imag(exp(−1/10 + j*π/5)*n)’

Time index n

Am

plitu

de

Return



Sinusoidal sequencesx [n] = Acos[!0n + �], 8n,where A is the amplitude, !0 the angular frequency, and � thephase of x [n]

0 5 10 15 20 25 30 35 40−1

0

1x[n] = cos[0.1 π n]

Time index n

Am

plitu

de

Return



Periodic sequencesx [n] periodic iff x [n] = x [n + N] 8nFundamental period: Smallest positive integer N that satisfies therelation.Sinusoidal sequences (A cos[!0n + �]) and complex exponentialsequences (B exp[|!0n]) are periodic sequences of period N if!0N = 2⇡k , where N and k are positive intergers.

Return


Discrete-time signals Classification of sequences

Classification of sequences

Symmetric sequences. Details

Signal energy. Details

Signal power. Details

Other types of classification Details



Symmetric sequencesA signal is conjugate symmetric (even if real) if, for all n,x [n] = x⇤[�n]A signal is conjugate antisymmetric (odd if real) if, for all n,x [n] = �x⇤[�n]Any signal can be decomposed into a sum of a conjugatesymmetric signal and a conjugate antisymmetric signal:

x [n] = xcs[n] + xca[n]where

xcs[n] =12(x [n] + x⇤[�n])

xca[n] =12(x [n]� x⇤[�n])

Return



Energy signals

Signal energy, Ex =P1

�1 x [n]x⇤[n] =P1

�1 |x [n]|2.

DefinitionSignal with finite energy, Ex < 1 is called a Energy signal.

A finite sample valued, infinite length sequence may or may nothave finite energy!A finite sample valued, finite length sequence have finite energy,but zero power.

Return



Power signals

Average signal power: Px = limL!11

2L+1PL

n=�L |x [n]|2.

DefinitionSignal with nonzero, finite average power is called a Power signal

Periodic signals are power signals (P = 1NPN�1

n=0 |x [n]|2).Return



Other types of classification

A sequence x [n] is said to be bounded if|x [n]| Bx < 1

A sequence x [n] is said to be absolutely summable ifP1n=�1 |x [n]| < 1

A sequence x [n] is said to be square-summable ifP1n=�1 |x [n]|2 < 1


Discrete-time signals Signal manipulations

Signal manipulations

Transformation of the independent variable:The index is n modified; y [n] = x [f [n]]where f [n] is some function of n.

I Time shifting: f [n] = n � n0, i.e. y [n] = x [n � n0].I Time reversal: f [n] = �n, i.e. y [n] = x [�n].I Time scaling: f [n] = Mn or f [n] = n/N, M,N 2 N .

F down-sampling: f [n] = Mn.Then y [n] = x [Mn], i.e. every Mth sample of x [n].

F up-sampling: f [n] = n/N.Then y [n] = x [f [n]] is defined as

y [n] =

(x⇥ n

N

⇤n = 0,±N,±2N, · · ·

0 otherwise.

Time shifting, Time reversal and Time scaling operations areorder-dependent, i.e. not commutative.


Discrete-time signals Signal manipulations

Signal manipulations ...

The most common types of amplitude transformations are straitforward and involves only point wise operations on the signal:

Addition (and in the same way subtraction)The sum of two signals is formed by pointwise addition of thesignal values: y [n] = x1[n] + x2[n], �1 < n < 1.Multiplication (and in the same way division)The multiplication of two signals is formed by point wise product ofthe signal values: y [n] = x1[n]x2[n], �1 < n < 1.ScalingAmplitude scaling of a signal by a constant c is accomplished bymultiplying every signal value by c: y [n] = cx [n], �1 < n < 1.

I This operation may also be considered to be the product of twosignals, x [n] and f [n] = c.


Discrete-time systems

Outline




Discrete-time systems Discrete-time systems

Discrete-time systems (again)Processes a given input sequence x [n] to generate anoutput sequence y [n].In most cases; a single-input, single output system

I Characterized through an input-output transformation:

y [n] = H{x [n]}

or x [n] �! H{·} �! y [n]

or x [n] H�! y [n].

Examples: Constant multiplier, unit delay, unit advance.I Visualized with a block diagram

I Example 2-input, 1-output systems: Adder, Signal multiplier(Modulator).


Discrete-time systems Discrete-time systems

Example (M-point moving-average system)

y [n] = 1MPM�1

k=0 x [n � k ]Used to smoothing random variations in data

Example (Accumulator)

y [n] =Xn

k=�1x [k ]

=Xn�1

k=�1x [k ] + x [n] = y [n � 1] + x [n]

(or) =X�1

k=�1x [k ] +

Xn

k=0x [n] = y [�1] +

Xn

k=0x [n], n � 0.

The term y [�1] is called the initial condition.


Discrete-time systems System properties

Classification of discrete-time systems

Classified intoI linear and nonlinear systems, orI time-varying and time-invariant systems, orI static and dynamic systems, orI causal and non-causal systems, orI stable and unstable systems, orI passive and lossless systems.

These are system properties



Causal, stable, static and passive systems

Causal system; the output at index n0 depends only in the input upto and including the index n0, and not on future values of the input.A system is bounded-input bounded-output (BIBO) stable if, forany input that is bounded, |x [n]| Mx < 1, the output will bebounded, |y [n]| My < 1.A system is static or memoryless if the output at any time n = n0depends only on the input at time n = n0.A system is passive if, for every finite-energy input x [n], the outputy [n] has, at most, the same enery, i.e.P1

n=�1 |y [n]|2 P1

n=�1 |x [n]|2 < 1.Lossless system iff above inequality is satisfied with an equal signfor every input.



System properties

AdditivityA system is said to be additive if

H{x1[n] + x2[n]} = H{x1[n]}+H{x2[n]}

for any signals x1[n] and x2[n].HomogeneityA system is said to be homogeneous if

H{cx [n]} = cH{x [n]}

for any complex constant c and for any input sequence x [n].I ) x [n] = 0 H

�! y [n] = 0, i.e. no new signals generated!



Linear systems

A discrete system is linear iffH{·} is both additive andhomogeneous, i.e. satisfiesthe general superpositionprinciple.i.e. H{a1x1[n] + a2x2[n]} =a1H{x1[n]}+ a2H{x2[n]}

To test if a system is linear, we test if it is both additive andhomogeneous.



Output of a linear systemAdditivity gives

I y [n] = H{x [n]} = H�P

1

k=�1x [k ]�[n � k ]

=P

1

k=�1H{x [k ]�[n � k ]}

Homogeneity givesI y [n] =

P1

k=�1H{x [k ]�[n � k ]} =

P1

k=�1x [k ]H{�[n � k ]}

If we define hk [n] to be the response to the system to a unitsample at time n = k ,

I hk [n] = H{�[n � k ]},we get that

I y [n] =P

1

k=�1x [k ]hk [n],

which is known as the superposition summation.



Time-invariant systems

A linear system T is time-invariant or shift-invariant iff the followingis true:x(n) �! H{·} �! y [n] �! Shift by k �!y [n � k ]

x(n)�! Shift by k �!x [n � k ] �! H{·} �! y [n � k ].



Linear time-invariant systems

A system that is both linear and shift-invariant is referred to as alinear time-invariant systemIt is denoted an LTI-system.

Output of a linear time-invariant systemIf h[n] is the response of a LTI system to the unit sample �[n], itsresponse to �[n � k ] will be h[n � k ].The superposition summation then becomes

I y [n] =P

1

k=�1x [k ]h[n � k ] = x [n] ⇤ h[n],

where ⇤ indicates the convolution operation.

The above equation is known as the convolution sum.



Impulse response and convolution sumThe response of a discrete-time system to a unit samplesequence, �[n], is denoted h[n] and called the unit sampleresponse or simply the impulse response.An LTI system is completely characterized in the time-domain byh[n], i.e. the response to the system to any input x [n] may befound once h[n] is known.Since the response of an LTI system to an input x [k ]�[n � k ] willbe x [k ]h[n � k ], the response, y [n] to an input

x [n] =X1

k=�1x [k ]�[n � k ]

will be

y [n] =X1

k=�1x [k ]h[n � k ]

=X1

k=�1x [n � k ]h[k ]



Convolution sum

The summationy [n] =

P1k=�1 x [k ]h[n � k ] =

P1k=�1 x [n � k ]h[k ]

is called the convolution sum of the sequences x [n] and h[n].Compact representation: y [n] = x [n] ⇤ h[n].Properties

I Commutative: x [n] ⇤ h[n] = h[n] ⇤ x [n]I Assosiative: {x [n] ⇤ h1[n]} ⇤ h2[n] = x [n] ⇤ {h1[n] ⇤ h2[n]}I Distributive: x [n] ⇤ {h1[n] + h2[n]} = x [n] ⇤ h1[n] + x [n] ⇤ h2[n]

Performing convolution:I Direct evaluationI Graphical approachI Slide rule method



Convolution sum; properties



Performing convolution; graphical approach



Stability and causuality of LTI systems

An LTI system is BIBO stable iff h[n] is absolutely summable, i.e.P1k=�1 |h[n]| < 1.

ExampleConsider h[n] = ↵nu[n].Then

P1k=�1 |↵nu[n]| =

P1k=0 |↵

n| = 1

1�|↵| if |↵| < 1Therefore, the system is BIBO stable for |↵| < 1.

An LTI system is causal iff the impulse response h[n] = 0, n < 0.

ExampleThe discrete-time accumulator is a causal system since it has a causalimpulse response given as h[n] =

Pnk=�1 �[k ] = u[n].


Discrete-time systems FIR and IIR LTI Systems

FIR and IIR LTI Systems

Causal FIR:h[n] = 0, n < 0 & n � M

Conv. formula:y [n] =

PM�1k=0 h[k ]x [n � k ]

FIR always stable since Mfinite.Causal IIR:y [n] =

P1k=0 h[k ]x [n � k ]

If h[n] = banu[n], |a| < 1,theny [n] =

P1k=0 banx [n � k ]

= ay [n � 1] + bx [n]IIR may be unstable.


Discrete-time systems Discrete-time systems described by difference equations

Linear coefficient difference-equation representation

An important subclass of LTI discrete-time systems ischaracterized by a linear constant coefficient difference equation.These are realizable in practice.h[n] = ↵nu[n] ) y [n] =

P1k=0 ↵

kx [n � k ] (1)(1) may be written asy [n] = ↵y [n � 1] + x [n] (2)(2) is a special case of a linear constant coefficient differenceequationThe general form:

y [n] =Pq

k=0 b[k ]x [n � k ]�Pp

k=1 a[k ]y [n � k ],where a[k ] and b[k ] are constants that defines the system.If one or more terms a(k) are nonzero; recursive systemIf all coefficients a(k) equal to zero; non-recursive system


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outline introduction to discrete time signals and systems

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