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**Unformatted text preview: **ECE3123 Digital Signal Processing
CHAPTER 2
DISCRETE-TIME SIGNALS &
SYSTEMS
Prof. Dr. Othman O. Khalifa
Electrical and Computer Engineering
Kulliyyah of Engineering
International Islamic University Malaysia Continuous-Time Sinusoidal Signals
A simple harmonic oscillation is mathematically
described by the following continuous-time sinusoidal
signal: The subcript a used with x(t) denotes an analog signal
which is characterized by; the sinusoid amplitude (A),
frequency (F) in cycles per second or hertz (Hz), and
phase (θ) in radians.
8/5/2009 Prof. Dr Othman O Khalifa ٢ ١ Discrete-Time Sinusoidal Signals
A discrete-time sinusoidal signal may be expressed as
x(n) = A cos(2πfn + θ ), −∞ < n < ∞ Use ω = 2πf to alternatively convert to radians per sample where n is the sample number, A is the sinusoid
amplitude, f is the frequency in cycles per sample, and θ
is the phase in radians. 8/5/2009 ٣ Classification of signals Multichannel
(signals are generated by multiple
sensors/sources which are measuring the
same parameter. Example: measurement of
ground acceleration a few kilometers from
epicenter of an earthquake, ECG etc)
Multidimentional
(The value of signal is a function of M
independent variables.)
8/5/2009 Prof. Dr Othman O Khalifa ٤ ٢ Continuous time Vs Discrete time (may
not be discrete valued)
Continuous valued Vs discrete valued
(quantized continuous valued discrete time
signal)
Deterministic Vs Random
(Deterministic, if signal can be uniquely
described by an explicit mathematical
expression e. g. sine, cosine, line, parabola
etc. Random signal examples noise,
unsynchronized digital signal etc )
8/5/2009 ٥ For Discrete time sinusoids
x(n) = Acos(ωn+φ); ω = 2πf;
f (cycles per sample) = F/Fs; Fs is the sampling freq Periodic, if ‘f’ is a rational number.
for periodicity, x(n+N) = x(n) for all n i.e.
Acos(2πf(n+N)+φ) = Acos(2πfn+φ)
– This relation is true if and only if there exists an
integer K such that 2πfN = 2πk i.e. f=k/N
– If k and N are relatively prime then N is called
the fundamental period of x[n].
– A small change in freq can results in large
change in period i.e. f1= 31/60 implies N1=60
but f2=30/60 implies N2= 2.
8/5/2009 Prof. Dr Othman O Khalifa ٦ ٣ Signal Types and Properties
Energy Vs Power signals
– Energy of x(n) = E= ∑-∞<n<∞|x(n)|2
<n<∞
– If E is finite, x(n) is called energy signal.
– Power of x(n) = P = lim(N tend to ∞) {1/(2N+1)}*
{∑- N<n<N|x(n)|2} If E is finite P=0 but if E is infinite P may or
may not be finite.
If P is finite then x(n) is called power signal.
8/5/2009 ٧ Energy Vs Power signals
Notes:
A signal is classified as energy type if its energy is finite
∞
(0<E<
)
Energy signals have Zero power
A signal is power type if its power is finite (0 < P < ∞)
N
1
P = lim
∑ x[n]
n →∞ 2 N + 1 n = − N Therefore,
8/5/2009 Prof. Dr Othman O Khalifa 2 1
EN
N →∞ 2 N + 1 P = lim ٨ ٤ Periodic Vs Aperiodic signals
– x(n+N) = x(n) for all n
– If no value of N satisfy it the signal is aperiodic. A sinusoid x(n)= Asin2πfn is periodic if f=k/N
Power of periodic signals =
(1/N)∑0<n<N--1|x(n)|2
0<n<N
8/5/2009 ٩ Periodic Vs Aperiodic signals Notes:
A periodic signal has no starting or
finishing time.
A periodic signal repeats endlessly.
A signal that dos not repeats it self is
siad to ne non-periodic or Aperiodic. 8/5/2009 Prof. Dr Othman O Khalifa ١٠ ٥ Symmetric (even) Vs Antisymmetric (odd)
– x(n) = x(-n); symmetric (even)
– x(n) = -x(-n); antisymmetric (odd) Any arbitrary signal can be expressed as
sum of two signal components, even and
odd. 8/5/2009 ١١ Examples 8/5/2009 Prof. Dr Othman O Khalifa ١٢ ٦ Classification of systems
1.
2.
3. Static (memoryless) Vs Dynamic (with memory)
(memoryless)
Time-invariant Vs Time Variant systems
TimeLinear Vs Nonlinear
(linear, must satisfy superposition principle, otherwise nonlinear.)
nonlinear.) 4. Causal Vs Noncausal
(o/p should depend upon present and past i/p but not on future i/p
{x(n+1)..}, otherwise noncausal. If signal is first recorded then
noncausal.
offline processing is done then only non causal systems are
possible to implement)
5 Stable Vs Unstable
(BIBO stable if every bounded i/p produces a bounded o/p,
o/p,
otherwise unstable.)
8/5/2009 ١٣ Static (memoryless) Vs Dynamic (with
memory)
Memoryless System
– A system is memoryless if the output y[n] at every value of n depends only on the
depends
input x[n] at the same value of n (static, if o/p depends upon present i/p only e.g. y(n)= nx(n)+bx3(n).
y(n)=
Dynamic, if depends upon past or present e.g. y(n)=x(n-1)+3x(n))
y(n)=x(nExample Memoryless Systems
– Square y[n] = (x[n]) 2 – Sign
Counter Example
– Ideal Delay System y[n] = sign{x[n]} y[n] = x[n − no ]
8/5/2009 Prof. Dr Othman O Khalifa ١٤ ٧ Linear Vs Nonlinear
Linear System: A system is linear if and only if T{x1[n] + x2[n]} = T{x1[n]} + T{x2[n]} (additivity)
and T{ax[n]} = aT{x[n]} (scaling) Examples
– Ideal Delay System y[n] = x[n − no ] T{x1[n] + x2[n]} = x1[n − no ] + x2[n − no ] T{x2 [n]} + T{x1[n]} = x1[n − no ] + x2[n − no ]
T{ax[n]}
=
ax1[n − no ]
aT{x[n]} = ax1[n − no ] 8/5/2009 ١٥ Time-invariant Vs Time Variant
systems (if i/p, x(n) results in y(n) then x(n-k) must results in y(n-k), otherwise system
i/p, x(n)
y(n)
x(ny(nis time variant. Ex: TI, y(n)=x(n)-x(n-1); TV, y(n)= x(n)*cos(wn))
y(n)=x(n)-x(ny(n)= x(n)*cos(wn)) y[n] = T{x[n]} ⇒ y[n − no ] = T{x[n − no ]} Time-Invariant (shift-invariant) Systems
Time(shift- – A time shift at the input causes corresponding time-shift at output
time- Delay the input the output is Example
– Square 2
y[n] = (x[n]) Delay the output gives y1 [ n ] = ( x[n]) 2 y [ n-n o ] = ( x[ n − no ]) 2 Counter Example
– Compressor System y[n] = x[Mn] 8/5/2009 Prof. Dr Othman O Khalifa Delay the input the output is
Delay the output gives y1 [ n ] = x[ Mn]
y [ n-n o ] = x ⎡ M ( n − no ) ⎤
⎣
⎦
١٦ ٨ Causal Vs Noncausal
Causality – A system is causal it’s output is a function of
it’
only the current and previous samples Examples – Backward Difference Counter Example – Forward Difference y[n] = x[n] − x[n − 1] y[n] = x[n + 1] + x[n] 8/5/2009 ١٧ Stable Vs Unstable
Stability (in the sense of bounded-input bounded-output BIBO)
boundedbounded– A system is stable if and only if every bounded input produces a
bounded output x[n] ≤ Bx < ∞ ⇒ y[n] ≤ By < ∞
2
Example
y[n] = (x[n])
– Square if input is bounded by x[n] ≤ B < ∞
x output is bounded by y[n] ≤ B2 < ∞
x Counter Example
– Log y[n] = log10 ( x[n] ) even if input is bounded by x[n] ≤ Bx < ∞ output not bounded for x[n] = 0 ⇒ y[0] = log10 ( x[n] ) = −∞ 8/5/2009 Prof. Dr Othman O Khalifa ١٨ ٩ Discrete-Time Signals: Time-Domain
Representation
Signals represented as sequences of
numbers, called samples
Sample value of a typical signal or sequence
denoted as x[n] with n being an integer in
the range ∞ ≤ ≤ ∞
x[n] defined only for integer values of n and
undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
8/5/2009 ١٩ Discrete-Time Signals: Sequences
Discrete-time signals are represented by sequence of numbers
Discrete– The nth number in the sequence is represented with x[n]
Often times sequences are obtained by sampling of continuous-time signals
continuous– In this case x[n] is value of the analog signal at xc(nT)
(nT)
– Where T is the sampling period
10
0
-10
0
10 t (ms) 20 40 60 80 100 10 20 30 40 50 n (samples) 0
-10
0
8/5/2009 Prof. Dr Othman O Khalifa ٢٠ ١٠ Basic Sequences and Operations
y[n] = x[n − no ]
1.5 Delaying (Shifting) a sequence ⎧0 n ≠ 0
δ[n] = ⎨
⎩1 n = 0
Unit sample (impulse) sequence
⎧0 n < 0
u[n] = ⎨
⎩1 n ≥ 0 1
0.5
0
-10 0 5 10 -5 0 5 10 -5 0 5 10
٢١ 1
0.5
0
-10 Unit step sequence -5 1.5 1 x[n] = Aαn
Exponential sequences 0.5 0
-10 8/5/2009 Discrete-Time Systems
Discrete-Time System is a mathematical operation that maps a given
DiscreteSystem
input sequence x[n] into an output sequence y[n] y[n] = T{x[n]} x[n] T{.} y[n] Example Discrete-Time Systems
Discrete– Moving (Running) Average y[n] = x[n] + x[n − 1] + x[n − 2] + x[n − 3]
– Maximum y[n] = max{x[n], x[n − 1], x[n − 2]}
– Ideal Delay System y[n] = x[n − no ]
8/5/2009 Prof. Dr Othman O Khalifa ٢٢ ١١ Discrete-Time Signals: TimeDomain Representation 8/5/2009 ٢٣ 8/5/2009 ٢٤ Prof. Dr Othman O Khalifa ١٢ 8/5/2009 ٢٥ Discrete-Time Signals: Time-Domain Representation Discrete-Time Signals are represented mathematically as sequences
Discreteof numbers.
The sequences of numbers x with nth number in the sequence is
denoted as x[n] where n being integer in the range of such as
[n]
n={…,-3,-2,-1,0,1,2,3,…}. It is also called time index.
… 3,- 2,- 1,0,1,2,3,…
n={
If the input signal to the systems is DTS, the output of the systems
systems
will be DTS.
INPUT
OUTPUT x[n]
[n]
8/5/2009 Prof. Dr Othman O Khalifa SYSTEM y[n]
[n]
٢٦ ١٣ Discrete-Time Signals: Time-Domain Representation
Components of DTS ::=> A delay y[n] = x[n-1]
[n]
[n=> An advance y[n] = x[n+1]
[n]
=> Scaling y[n] = ax[n]
[n]
[n]
=> Downsampling (Decimation) y[n] = x[nM]
[n]
[nM]
=> Upsampling (Interpolation) y[n] = x[n/L]
[n]
[n/L]
=> Nonlinear y[n] = tanh[ax(n)]
[n] tanh[ax (n)]
8/5/2009 ٢٧ Discrete-Time Signals: Time-Domain Representation A DTS component is visualized below: 8/5/2009 Prof. Dr Othman O Khalifa ٢٨ ١٤ GRAPHICAL REPRESENTATION OF DTS Graphical Representation of a DTS is
shown below: (infinite) 8/5/2009 ٢٩ Discrete-Time Signals: Time-Domain Representation The discrete-time signal is obtained by performing periodic
discretesampling of analog signal. The sampling interval or period is
denoted as Ts. Thus the sampling Frequency can be defined
as reciprocal of Ts, namely, Fs = 1 / Ts.
When the analog is sampled at certain period of time, the
discrete-time signal can be written as below :discrete:- x[n] = xa[t] = xa[nTs],
[n]
[t] 8/5/2009 Prof. Dr Othman O Khalifa n = …,-2,-1,0,1,2,...
2,- ٣٠ ١٥ Discrete-Time Signals: Time-Domain Representation
When the sinusoidal analog signal being sampled to become a
discrete signal (digital signal), the frequency of input signal will
be changed to digital frequency as illustrated below.
Given a signal to the system is defined as:
xa[t] = Acos(ωt + Ф) = Acos(2πFt + Ф)
[t] Acos(
Acos(2π
The signal being sampled at certain period of time, Ts (Fs =1 / Ts).
Thus the form of the discrete (digital signal) now become:
xa[t] = x[n] = Acos(ωt + Ф) = Acos(2πFnT + Ф),
t => nT,
[t]
[n] Acos(
Acos(2π
nT,
x[n] = Acos(2πFnT + Ф) = Acos(2πnF/FS + Ф)
[n] Acos(2π
Acos(2π nF/F
Hence, the digital Frequency is defined as :
f = F/FS
where F is Bandwidth input signal frequency & Fs is the
Sampling Frequency and n is a time index.
8/5/2009 ٣١ Discrete-Time Signals: Time-Domain Representation
Example :
The input signal to the system is continuous cosine signal with
Amplitude of 10 and signal frequency of 40Hz. The signal is sampled at
sampled
Sampling Frequency, Fs of 400Hz with sequence index, n ranging from 10 to 10.
Steps :
1. Determine the Input Signal, xa[t] = 10cos(2*π*40*t)
[t] 10cos(2*π
since fo = 40Hz with zero phase shift (Ф = 0)
2. Calculate the Sampling Period, Ts = 1/Fs = 1/400 = 0.0025s
3. Determine the Discrete signal,
x[n] = xa[t] = xa[nTs] = xa[n*0.0025]
[n]
[t]
[n*0.0025]
=> 10cos(2*pi*40*n*0.0025), n = -10 to 10 EX.
4. Use MATLAB to compile the program and plot the graph
8/5/2009 Prof. Dr Othman O Khalifa ٣٢ ١٦ GRAPHICAL REPRESENTATION OF DISCRETEDISCRETETIME SIGNAL at TS = 0.0025s 8/5/2009 ٣٣ SEQUENCE REPRESENTATION
There are several type of sequences in Discrete-Time Signals
Discretesuch as unit sample or unit impulse, unit step, Real&complex
exponential and Sinusoidal. The unit impulse is defined as :
a. Unit Impulse => δ[n] = {1, n = 0;
0, n ≠ 0 }
Graphical representation of Unit Impulse : 8/5/2009 Prof. Dr Othman O Khalifa ٣٤ ١٧ SEQUENCE REPRESENTATION
b. Unit Step => u[n] = {1, n ≥ 0;
[n]
0, n < 0 }
Graphical representation of Unit Step : 8/5/2009 ٣٥ SEQUENCE REPRESENTATION c. 8/5/2009 Prof. Dr Othman O Khalifa Real exponential => x[n] = Aаn; A is constant and а is a
[n] Aа
real number.
Graphical Representation of Real exponential : ٣٦ ١٨ SEQUENCE REPRESENTATION
d. Complex exponential => x[n] = Aеjωn; ω frequency of
[n] Aе
complex exponential sinusoid, A is a constant
Graphical representation of Complex exponential : 8/5/2009 ٣٧ SEQUENCE REPRESENTATION
e. Sinusoidal 8/5/2009 Prof. Dr Othman O Khalifa ٣٨ ١٩ SEQUENCE REPRESENTATION
e. The Unit Impulse can be shifted or delayed. The shifted Unit Impulse is
Impulse
denoted as ::δ[n-k] => The unit impulse is shifted to right by k
δ[n+k] => The unit impulse is shifted to left by k
n+k] Example :
The sequence, p[n] is expressed as :
p[n]
p[n] = a-3 δ[n + 3] + a1 δ[n – 1] + a2 δ[n – 2] + a7 δ[n – 7]
p[n]
Graphical representation of Shifted unit Impulse of p[n]:
p[n]: 8/5/2009 ٣٩ SEQUENCE REPRESENTATION
The sequence of Discrete-Time Signals can be represented in
Discreteterm of Shifted Unit impulse as defined below :
∞ x[n] = Σx[k]δ[n-k]
[n]
[k]
k=-∞
k=- The unit step sequence can be defined in term of Shifted Unit
Impulse as shown below :
∞ u[n] =Σδ[n-k]
[n] =Σ
k= 0 8/5/2009 Prof. Dr Othman O Khalifa ٤٠ ٢٠ SEQUENCE REPRESENTATION
The unit impulse can be defined in term of Shifted Unit Step
as shown below :
δ[n] = u[n]-u[n-1]
[n]- [n- 8/5/2009 ٤١ Introduction to LTI System
Discrete-time Systems
– Function: to process a given input
sequence to generate an output
sequence
x[n]
Input
sequence Discrete-time system y[n]
Output
sequence Fig: Example of a single-input, single-output system
8/5/2009 Prof. Dr Othman O Khalifa ٤٢ ٢١ Review
Classification of Discrete-time System
System with and without memory
Causality
Stability
Time-invariance
Linearity 8/5/2009 ٤٣ Classification of Discrete-time System
Memoryless system
– Output at a given time is dependent on
the input at only that same time y[n] = (2 x[n] − x 2 [n]) 2 Identity system
y (t ) = x(t ) or y[n] = x[n]
8/5/2009 Prof. Dr Othman O Khalifa ٤٤ ٢٢ Classification of Discrete-time System
System with memory
– Summer
n y[n] = ∑ x[k ] k =−∞ – Delay y[ n] = x[n − 1]
8/5/2009 ٤٥ Invertible
– distinct inputs lead to distinct output
– an inverse system exists Example
y (t ) = 2 x(t )
→ inverse system is w(t ) =
8/5/2009 Prof. Dr Othman O Khalifa 1
y (t )
2
٤٦ ٢٣ Classification of Discrete-time System
Example y[n] = n ∑ x[k ] k =−∞ → inverse system is w[n] = y[n] - y[n -1] 8/5/2009 ٤٧ Classification of Discrete-time System 8/5/2009 Prof. Dr Othman O Khalifa ٤٨ ٢٤ Classification of Discrete-time System
Causal
– The output at any time depends on
values of the input at only the present
and past times
– Example: memory system, RC circuit, …,
etc. Noncausal y[n] = x[n] − x[n + 1]
y (t ) = x(t + 1) 8/5/2009 ٤٩ Classification of Discrete-time System
Example causal
⎧n > 0,
(1) y[n] = x[−n] ⇒ ⎨
⎩ n < 0, noncausal (2) y (t ) = x(t ) cos(t + 1)
→ y (t ) = x(t ) g (t ) where g (t ) = cos(t + 1)
→ causal system
8/5/2009 Prof. Dr Othman O Khalifa ٥٠ ٢٥ Classification of Discrete-time System
Stable system
– If the input is bounded then the output
must be bounded and cannot diverge
– Example: RC circuit … etc Unstable system
– Example: y[n] = n ∑ u[k ] = (n + 1)u[n] k =−∞ 8/5/2009 ٥١ Classification of Discrete-time System
Example
– Look for a specific bounded input that
leads to an unbounded output (1) S1 : y (t ) = tx(t )
<ans> if we consider x(t ) = 1
→ y (t ) = t - - - -unstable
8/5/2009 Prof. Dr Othman O Khalifa ٥٢ ٢٦ Classification of Discrete-time System
Example
<ans> (2) S 2 : y (t ) = e x (t ) let B be an arbitrary positive number ,
and let x(t ) be an arbitrary signal bounded by B
i.e. − B < x(t ) < B
→ e − B < y (t ) < e B − − − − stable
8/5/2009 ٥٣ Classification of Discrete-time System
Time-invariant
– The behavior and characteristics of the
system are fixed over time
– Example: RC circuit of Fig1.1 is timeinvariant if the resistance and capacitance
values are constant
– If a time-shift in the input signal results in
an identical time shift in the output signal
8/5/2009 Prof. Dr Othman O Khalifa ٥٤ ٢٧ Classification of Discrete-time System
Example y (t ) = sin[ x(t )] <ans> let y1 (t ) = sin[ x1 (t )]
and x2 (t ) = x1 (t − t0 )
→ y2 (t ) = sin[ x2 (t )] = sin[ x1 (t − t0 )] − −(a )
and y1 (t − t0 ) = sin[ x1 (t − t0 )] − − − − − (b) Q (a ) = (b) ∴ time − in var iance
8/5/2009 ٥٥ Classification of Discrete-time System
Example y[n] = nx[n]
<ans>
let y1[n] = nx1[n]
and x2 [n] = x1[n − n0 ]
→ y2 [n] = nx2 [n] = nx1[n − n0 ] − − − −(a )
and y1[n − n0 ] = (n − n0 ) x1[n − n0 ] − −(b)
8/5/2009 Prof. Dr Othman O Khalifa Q (a ) ≠ (b) ∴ time - var iance ٥٦ ٢٨ Classification of Discrete-time System
Linear
– Additivity property The response to x1 (t ) + x2 (t ) is y1 (t ) + y2 (t )
– Scaling (homogeneity) property The response to ax1 (t ) is ay1 (t ),
where a is any complex cons tan t
8/5/2009 ٥٧ Classification of Discrete-time System
Combine the two properties
continuous : ax1 (t ) + bx2 (t ) → ay1 (t ) + by2 (t )
discrete : ax1[n] + bx2 [n] → ay1[n] + by2 [n] where a and b are complex cons tan ts Superposition property
– An zero input results in an zero output 0 = 0 ⋅ x[n] → 0 ⋅ y[n] = 0
8/5/2009 Prof. Dr Othman O Khalifa ٥٨ ٢٩ Classification of Discrete-time System
Example y (t ) = tx(t )
<ans>
x1 (t ) → y1 (t ) = tx1 (t )
x2 (t ) → y2 (t ) = tx2 (t )
let x3 (t ) = ax1 (t ) + bx2 (t ), where a and b are arbitrary scalar
→ y3 (t ) = tx3 (t ) = t (ax1 (t ) + bx2 (t ))
= atx1 (t ) + btx2 (t ) = ay1 (t ) + by2 (t )
→ linear 8/5/2009 ٥٩ Classification of Discrete-time System
Example
y[n] = 2 x[n] + 3
<ans> if x[n] = 0 → y[n] = 3 ≠ 0
→ nonlinear 8/5/2009 Prof. Dr Othman O Khalifa ٦٠ ٣٠ Classification of Discrete-time System
Incrementally linear system
– Difference between the responses to any
two outputs to an incrementally linear
system is a linear function of the
difference between the two inputs
– Example:
y[n] = 2 x[n] + 3
→ y1[n] − y2 [n] = {2 x1[n] + 3} − {2 x2 [ n] + 3}
8/5/2009 = 2{x1[n] − x2 [n]} ٦١ Classification of Discrete-time System Linear System
– Most widely used
– A Discrete-time system is a linear system if the
Discretesuperposition principle always hold.
– If y1[n] and y2[n] are the response to the input
sequences x1[n] and x2[n], then
x[n]
= αx1[n] + βx2[n]
8/5/2009 Prof. Dr Othman O Khalifa Linear DTS y[n]
= αy1[n] + βy2[n]
٦٢ ٣١ Classification of Discrete-time System
Example Is the system described below linear or not ?
y[n] = x[n] + x[n-1]
[n]
[n]
[n- Step : a. Now, applying superposition by considering input as :
x[n] = ax[n] + bx[n]
[n]
[n]
[n]
b. Substitute the equation above with equation in (a), become
y[n] = (ax[n] + bx[n]) + (ax[n-1] + bx[n-1])
[n] (a [n]
[n])
(ax [nbx [nc. Rearrange the equation above become ::y[n] = a(x[n] + x[n-1]) + b(x[n] + x[n-1]) => ay[n] + by[n]
[n[n]
[n]
[n] a(x [n]
[nb(x [n]
c. The system is Linear since superposition is hold. 8/5/2009 ٦٣ Classification of Discrete-time System
Shift-invariant System/Time-Invariant
System
– A shift (delay) in the input sequence cause a
shift (shift) to the output sequence
– If y1[n] is the response to an input x1[n], then
the response to an input x[n] = x1[n - no] is
y[n] = y1[n - no] 8/5/2009 Prof. Dr Othman O Khalifa ٦٤ ٣٢ Classification of Discrete-time System
Causal System
– Changes in output samples do not
precede changes in input samples
– y[no] depends only on x[n] for n ≤ no
– Example: y[n] = x[n]-x[n-1] 8/5/2009 ٦٥ Classification of Discrete-time System
Stable System
– For every bounded input, the output is
also bounded (BIBO)
– Is the y[n] is the response to x[n], and if
|x[n]| < Bx for all value of n
then
|y[n]| < By for all value of n
Where Bx and By are finite positive constant
8/5/2009 Prof. Dr Othman O Khalifa ٦٦ ٣٣ Impulse and Step Response
If the input to the DTS system is Unit
Impulse (δ[n]), then output of the
system will be
Impulse Response (h[n]). If the input to the DTS system is Unit
Step (μ[n]), then output of the system
will be
Step Response (s[n]).
8/5/2009 ٦٧ Input-output Relationship
A Linear time-invariant system satisfied both
the linearity and time invariance properties.
An LTI discrete-time system is charact...

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- Summer '10
- ProfOthmanOKhalifa
- Digital Signal Processing, Signal Processing, LTI system theory, Prof. Dr Othman O Khalifa, Othman O Khalifa