discrete-time signals and systems

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PSD, Discrete-Time Systems

DISCRETE-TIME SIGNALS and SYSTEMS

Sub-topics:Discrete-Time Signals (DTS)

-. Basic DTS-. Classification of DTS-. Simple Manipulation of DTS

Discrete-Time Systems-. Input-Output Description of Systems-. Classification of DT Systems-. Interconnection of DT Systems

Implementation of Discrete-Time SystemsCorrelation of Discrete-Time Signals

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Discrete-Time Signals

• A DTS x(n) is a function of an independent variable that is integer

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Some Representations of DTS

1. Functional representation

2. Tabular representation

3. Sequence representation

lainnya ,0

2n ,4

3,1n ,1

)n(x

n … -2 -1 0 1 2 3 4 5 …

x(n) … 0 0 0 1 4 1 0 0 …

x(n) = { …, 0,0,1,4,1,0,0, …} .............................................................................................. (3.2)

x(n) = {0,1,4,1,0,0, …} ........................................................................................................ (3.3)

x(n) = {3,-1,-2,5,0,4,-1} ....................................................................................................... (3.4)

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Some Elementery DTS

1. The Unit Sample Sequence or Unit Impulse

1. The Unit Step Signal

0n ,0

0n ,1)n(

……

1

(n)

n

0n ,0

0n ,1)n(u

……

u(n)

1

n

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• 3. The Unit Ramp Signal

4. The Exponential Signal

0n ,0

0n ,n)n(ur ……

ur(n)

n

x(n) = an,

x(n) = rn ejn = rn (cosn + j sinn)

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• Classification of DTS– Energy signals and power signals

– Energy signal E can be finite and infinite.– If E is finite (0<E<∞), then x(n) is energy signal, P=0.– If E is infinite, then P can be finite or infinite.– If P is finite and P0, then signal x(n) is called power signal.

– Periodic signals and aperiodic signalsx(n+N) = x(n), all n -> periodic (N = period)

Otherwise is aperiodic

n

2)n(xE

N

Nn

2

N)n(x

1N2

1limP

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– Symmetric (even) and anti-symmetric (odd) signals

• x(-n) = x(n)• x(-n) = -x(n)

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• Manipulation of Discrete-Time Signals

• Transformation of the independent variable (time)

– A signal x(n) is shifted in time by replacing the independent variable n by n-k, where k is integer

– Results: delay of the signal (k is positive) or an advance of the signal (k is negative)

– Folding or Reflection of signal (n becomes –n about the time origin n = 0)

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• Folding and shifting process

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• Downsampling process• Replacing n by n, where is integer

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• Addition, multiplication, and scaling of sequences

• Amplitude scaling: y(n) = A x(n) ; -∞<n<∞ ; A is a constant• The sum of two signals: y(n) = x1(n) + x2(n); -∞<n<∞ • The product of two signals: y(n) = x1(n).x2(n); -∞<n<∞

• Discrete-Time Systems• y(n) [x(n)]

n

k

1n

k)n(x)1n(y)n(x)k(x)k(x)n(y

Accumulator: the system computes the current value of the input to the previous output value

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• Block Diagram Representation of Discrete-Time Systems

• An Adder: memoryless process• A constant multiplier: memoryless process• A signal multiplier: memoryless process• A unit delay element: Z-1 is not memoryless• A unit advance element: not memoryless

+

x1(n)

x2(n)

y(n) = x1(n) + x2(n) ax(n) y(n) = a x(n)

xx1(n)

x2(n)

y(n) = x1(n) x2(n)

Z-1x(n) y(n) = x(n-1)

Zx(n) y(n) =

x(n+1)

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• Classification of Discrete-Time Systems• Static versus dynamic systems

– Static» It is memoryless» Its output at any instant n depends at most on the input

sample at the same time, but not on past or future samples of the input.

– Dynamic» It has a memory» Its output at time n is completely determined by the input

samples in the interval from n-k to n(k0), the system is said to have memory of duration k.

» If k=0, the system is static» If 0<k<, the system is said to have finite memory» If k = , the system is said to have infinite memory

• Time-invariant versus time-variant systems• Linear versus nonlinear systems• Causal versus non-causal systems• Stable versus unstable systems

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• Static vs Dynamic systems

y(n) = a x(n) dan y(n) = n x(n) + b x3(n)

Y(n) = x(n) + 3 x(n-1)

0k

n

0k

)kn(x)n(y

)kn(x)n(y

• Time-invariant versus time-variant systems• TI systems or shift invariant

– If its input-output characteristics do not change with time

Theorem. A relaxed system is time invariant if and only if:

)n(y)n(x )kn(y)kn(x

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• Time-variant– If the output

y(n,k) y(n-k), even for one value of k.

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• Linear vs non-linear systems• A linear system is a system that satisfies the superposition

principle• Otherwise non-linear systems

Theorem. A system is linear if and only if

)]n(x[a)]n(x[a)]n(xa)n(xa[ 22112211

E.g. Linear systems:

y1(n) = x1(n2) dan y2(n) = x2(n2), with superposition principle:

y3(n) = [a1x1(n) + a2x2(n)] = a1x1(n2) + a2x2(n2), then

a1y1(n) + a2y2(n) = a1x1(n2) + a2x2(n2)

e.g. Non-Linear Systems:y(n) = ex(n), jika x(n) = 0 then y(n) = 1.

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• Causal versus non-causal systems– Causal system

• If the output of the system at any time n [i.e., y(n)] depends only on present and past inputs [i.e., x(n), x(n-1), x(n-2), …]

– Non-causal systems• Its output depends not only on present and past inputs but

also on future inputs

y(n) = x(n) – x(n-1); y(n) = a x(n)

y(n) = x(n) + 3 x(n+4); y(n) = x(n2); y(n) = x(2n)

• Stable versus unstable systems– Stable system

• Theorem. An arbitrary relaxed system is said to be bounded input – bounded output (BIBO) stable if and only if every bounded input produces a bounded output.

• Bounded: existed finite numbers

|x(n)| Mx < ∞ ; |y(n)| My < ∞ ;

n)n(h

x(n), y(n) = input, output

Mx, My = finite number

h(n) = impuls respon

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• Unstable system• If, for some bounded input sequence x(n), the output is unbounded

(infinite)

h(n) = -(0,5)n u(-n-1)

h(n) = 2n u(n)

• Interconnection of Discrete-Time Systems• Cascade interconnection/series

• Parallel interconnection

1 2

y(n)x(n) y1(n)

c

y1(n) = 1 [x(n)]

y(n) = 2 [y1(n)] = 2 [1 [x(n)]]

c 2 1

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

Cascade interconnection

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• Parallel:

• y3(n) = y1(n) + y2(n) = 1[x(n)] + 2[x(n)] = (1+ 2)[x(n)] = p[x(n)]

p = (1+ 2)

1

2

y3(n)x(n)

y1(n)

+

y2(n)

p

• Analysis of Discrete-Time Linear Time-Invariant Systems: – Convolution technique

• It involves input, output signals, and impuls respons.• Math. Techniques that combine two signals to create a new signal

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• Examples of application of convolution

a. Low-Pass Filter (LPF)

b. High-Pass Filter (HPF)

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• In math. Expression:

• y(n): output signal, x(n): input signal and h(n): impuls respon

• Convolution can be done in 4 steps:– Folding: Fold h(k) to k=0 to get h(-k)– Shifting: shift h(-k) by n0 to the right (left) if n0 is

positive (negative) to get h(n0 – k)– Multiplication: multiply x(k) to h(n0 – k) to obtain the

sequence vno(k) x(k)h(no – k).– Summation: Sum the sequence vno(k) to obtain the

output value at n = no.

k)kn(h)k(x)n(h*)n(x)n(y

k)kn(x)k(h)n(x*)n(h)n(y

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Convolution:

vo(k) x(k)h(-k)

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• Property of convolution and the interconnection of LTI systems

• Commutative Law: x(n) * h(n) = h(n) * x(n)

• Associative Law: [x(n) * h1(n)] * h2(n) = x(n) * [h1(n) * h2(n)]

• Distributive Law:x(n) * [h1(n) + h2(n)] = x(n) * h1(n) + x(n) * h2(n)

– Commutative Law

– Associative Law

h(n)x(n) y(n)

x(n)y(n)h(n)

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• Distributive Law

• Interconnection of LTI systems– Direct-Form I and Direct-Form II.

• Direct-Form I uses delay (memory) element seperately between sample of input signal and output signal.

• Direct-Form II, both input and output signal use the same delay elements. Hence Direct-Form II is more eficient.

y(n) = -a1 y(n-1) + bo x(n) + b1 x(n-1)

v(n) = bo x(n) + b1 x(n-1) (non-recursive) ; y(n) = -a1 y(n-1) + v(n) (Fig. b)

or

w(n) = -a1 w(n-1) + x(n) y(n) = bo w(n) + b1 w(n - 1)

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• Direct-Form I and Direct-Form II

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• Correlation technique in Discrete Systems– It is used to measure the degree to which the two signals are similar and

thus to extract some information that depends to a large extent on the application

– Cross-correlation: correlation technique on two different signals

– Autocorrelation: on two same signals

The relationship between transmitted signal and reflected signal [ x(n) and y(n)]

y(n) = x(n – D) + w(n)

= attenuation factor in the round-trip transmission

D = delay round-trip

w(n) = additive noise system

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• Crosscorrelation

,...2,1,0l)ln(x)n(y)l(ryx

,...2,1,0l)ln(x)n(y)l(rn

xy

)l(r)l(r yxxy

,...2,1,0l)ln(x)n(x)l(rn

xx

• Autocorrelation

Correlation technique of sequences:

a. Shifting of any one of sequences.

b. Multiplication of two sequences.

c. Summing of all values of n.

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• Normalized crosscorrelation dan autocorrelation:

)0(r)0(r

)l(r)l(

yyxx

xyxy

)0(r

)l(r)l(

xx

xxxx

discrete-time signals and systems

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