Transcript ADC, DAC, AND DISCRETE

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 Systems
Correlation of Discrete-Time Signals
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PSD, Discrete-Time Systems
Discrete-Time Signals
• A DTS x(n) is a function of an independent
variable that is integer
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PSD, Discrete-Time Systems
Some Representations of DTS
1. Functional representation
 1, n  1,3

x( n )   4 , n  2
0 , lainnya

2. Tabular representation
n
…
-2
-1
0
1
2
3
4
5
…
x(n)
…
0
0
0
1
4
1
0
0
…
3. Sequence representation
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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PSD, Discrete-Time Systems
Some Elementery DTS
1. The Unit Sample Sequence or Unit Impulse
(n)
1, n  0
( n )  
0 , n  0
1
…
1. The Unit Step Signal
…
n
1, n  0
u( n )  
0 , n  0
u(n)
1
…
…
n
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PSD, Discrete-Time Systems
• 3. The Unit Ramp Signal
ur(n)
n , n  0
ur ( n )  
0 , n  0
…
…
4. The Exponential Signal
n
x(n) = an,
x(n) = rn ejn = rn (cosn + j sinn)
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PSD, Discrete-Time Systems
• Classification of DTS
– Energy signals and power signals
E

 x( n )
n  
–
–
–
–
2
N
1
2
P  lim
 x( n )
N  2 N  1 n   N
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 P0, then signal x(n) is called power signal.
– Periodic signals and aperiodic signals
x(n+N) = x(n), all n -> periodic (N = period)
Otherwise is aperiodic
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PSD, Discrete-Time Systems
– Symmetric
(even) and antisymmetric (odd)
signals
• x(-n) = x(n)
• x(-n) = -x(n)
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PSD, Discrete-Time Systems
• Manipulation of DiscreteTime 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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PSD, Discrete-Time Systems
• Folding and shifting process
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PSD, Discrete-Time Systems
• Downsampling process
• Replacing n by n, where  is integer
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PSD, Discrete-Time Systems
• 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)]
Accumulator: the system computes the current value of the input to the previous
output value
n
n 1
y( n )   x( k )   x( k )  x( n )  y( n  1 )  x( n )
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k  
k  
PSD, Discrete-Time Systems
• Block Diagram Representation of DiscreteTime 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)
x(n)
y(n) = x1(n) + x2(n)
a
y(n) = a x(n)
+
x2(n)
y(n) = x1(n) x2(n)
x1(n)
x
x(n)
Z-1
x(n)
x2(n)
y(n) = x(n-1)
Z
y(n) =
x(n+1)
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PSD, Discrete-Time Systems
• 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(k0), 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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PSD, Discrete-Time Systems
• Static vs Dynamic systems
y( n ) 
3
y(n) = a x(n) dan y(n) = n x(n) + b x (n)
y( n ) 
Y(n) = x(n) + 3 x(n-1)
n
 x( n  k )
k 0

 x( n  k )
k 0
• 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:

x( n )  y( n )
x( n  k )

 y( n  k )
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PSD, Discrete-Time Systems
• Time-variant
– If the output
y(n,k)  y(nk), even for
one value of
k.
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PSD, Discrete-Time Systems
• 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
 [ a1 x1( n )  a2 x2 ( n )]  a1 [ x1( n )]  a2 [ x2 ( n )]
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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PSD, Discrete-Time Systems
• 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), y(n) = input, output
|x(n)|  Mx < ∞ ; |y(n)|  My < ∞ ;

 h( n )  
n  
Mx, My = finite number
h(n) = impuls respon
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PSD, Discrete-Time Systems
• 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
x(n)
1
y1(n)
2
y(n)
c
y1(n) = 1 [x(n)]
Cascade interconnection
y(n) = 2 [y1(n)] = 2 [1 [x(n)]]
c 2 1
y(n) = c [x(n)]
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PSD, Discrete-Time Systems
• Parallel:
• y3(n) = y1(n) + y2(n) = 1[x(n)] + 2[x(n)] = (1+ 2)[x(n)] = p[x(n)]
• p = (1+ 2)
p
1
y1(n)
x(n)
y3(n)
+
2
y2(n)
• 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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PSD, Discrete-Time Systems
• Examples of application of convolution
a. Low-Pass Filter (LPF)
b. High-Pass Filter (HPF)
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PSD, Discrete-Time Systems
• In math. Expression:
y( n )  x( n )* h( n ) 

 x( k )h( n  k )
y( n )  h( n )* x( n ) 
k  

 h( k )x( n  k )
k  
• 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.
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PSD, Discrete-Time Systems
Convolution:
vo(k)  x(k)h(-k)
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PSD, Discrete-Time Systems
• 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
x(n)
y(n)
h(n)
h(n)
x(n)
y(n)
– Associative Law
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PSD, Discrete-Time Systems
• 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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PSD, Discrete-Time Systems
• 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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PSD, Discrete-Time Systems
• Crosscorrelation
ryx ( l ) 
rxy ( l ) 

 y( n )x( n  l )
l  0 ,1,2 ,...


 y( n )x( n  l )
rxy ( l )  ryx ( l )
l  0 ,1,2 ,...
n  
• Autocorrelation
rxx ( l ) 

 x( n )x( n  l )
l  0 ,1,2 ,...
n  
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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PSD, Discrete-Time Systems
• Normalized crosscorrelation dan autocorrelation:
 xy ( l ) 
rxy ( l )
rxx ( 0 )ryy ( 0 )
rxx ( l )
 xx ( l ) 
rxx ( 0 )
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