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

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

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

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

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

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

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

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)

Folding and shifting process

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

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

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)

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

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:

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

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.

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 < ∞ ;

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

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

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

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.

Convolution: vo(k)  x(k)h(-k)

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)

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)

Direct-Form I and Direct-Form II

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

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.

Normalized crosscorrelation dan autocorrelation: