1. 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

2. Discrete-Time Signals
A DTS x(n) is a function of an independent variable that is integer

3. Some Representations of DTS
Functional representation
2. Tabular representation
3. Sequence representation n … -2 -1 1 2 3 4 5
x(n)

4. … … Some Elementery DTS The Unit Sample Sequence or Unit Impulse
The Unit Step Signal … 1
(n) n …
u(n) 1 n

5. 4. The Exponential Signal
3. The Unit Ramp Signal
4. The Exponential Signal …
ur(n) n
x(n) = an,

6. 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 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

7. Symmetric (even) and anti-symmetric (odd) signals
x(-n) = x(n)
x(-n) = -x(n)

8. 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)

9. Folding and shifting process

10. Downsampling process
Replacing n by n, where  is integer

11. 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

12. 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) a
x(n)
y(n) = a x(n) x
x1(n)
x2(n)
y(n) = x1(n) x2(n)
Z-1
x(n)
y(n) = x(n-1) Z
x(n)
y(n) = x(n+1)

13. 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

14. Static vs Dynamic systems
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:

15. Time-variant
If the output y(n,k)  y(n-k), even for one value of k.

16. 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
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.

17. 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
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
Mx, My = finite number
h(n) = impuls respon
|x(n)|  Mx < ∞ ; |y(n)|  My < ∞ ;

18. Interconnection of Discrete-Time Systems
Unstable system
If, for some bounded input sequence x(n), the output is unbounded (infinite)
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

19. Analysis of Discrete-Time Linear Time-Invariant Systems:
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

20. Examples of application of convolution
a. Low-Pass Filter (LPF)
b. High-Pass Filter (HPF)

21. Convolution can be done in 4 steps:
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.

22. Convolution:
vo(k)  x(k)h(-k)

23. 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)

24. Interconnection of LTI systems Direct-Form I and Direct-Form II.
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)

25. Direct-Form I and Direct-Form II

26. 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

27. Crosscorrelation Autocorrelation Correlation technique of sequences:
a. Shifting of any one of sequences.
b. Multiplication of two sequences.
c. Summing of all values of n.

28. Normalized crosscorrelation dan autocorrelation: