Digital Signal Processing
Discrete-Time Signals and Systems (1)
Moslem Amiri, V´aclav Pˇrenosil
Embedded Systems Laboratory
Faculty of Informatics, Masaryk University
Brno, Czech Republic
amiri@mail.muni.cz
prenosil@fi.muni.cz
March, 2012
Discrete-Time Signals
A discrete-time signal x(n) (≡ xa(nT)) is a function of an independent
variable that is an integer
We assume that x(n) is defined for every n for −∞ < n < ∞
x(n) is not defined for non-integer values of n
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Some Elementary Discrete-Time Signals
Unit sample sequence or unit impulse
δ(n) ≡
1, for n = 0
0, for n = 0
Figure 1: Graphical representation of the unit sample signal.
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Some Elementary Discrete-Time Signals
Unit step signal
u(n) ≡
1, for n ≥ 0
0, for n < 0
Figure 2: Graphical representation of the unit step signal.
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Some Elementary Discrete-Time Signals
Unit ramp signal
ur (n) ≡
n, for n ≥ 0
0, for n < 0
Figure 3: Graphical representation of the unit ramp signal.
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Some Elementary Discrete-Time Signals
Exponential signal
x(n) = an for all n
If a is real, x(n) is a real signal
Figure 4: Graphical representation of exponential signals.
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Some Elementary Discrete-Time Signals
Exponential signal
x(n) = an for all n
If a is complex
a ≡ rejθ
x(n) = rn
ejθn
= rn
(cos θn + j sin θn)
x(n) can be represented by separately plotting real part and imaginary
part as functions of n
xR (n) ≡ rn
cos θn
xl (n) ≡ rn
sin θn
Alternatively, x(n) can be represented by separately plotting amplitude
and phase functions
|x(n)| = A(n) ≡ rn
∠x(n) = φ(n) ≡ θn
By convention, φ(n) is plotted over −π < θ ≤ π or 0 ≤ θ < 2π
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Some Elementary Discrete-Time Signals
Figure 5: Graph of the real (xR (n) ≡ rn
cos θn) and imaginary (xl (n) ≡ rn
sin θn)
components of a complex-valued exponential signal for r = 0.9 and θ = π/10.
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Some Elementary Discrete-Time Signals
Figure 6: Graph of amplitude and phase function of a complex-valued
exponential signal: (a) graph of A(n) = rn
, r = 0.9; (b) graph of
φ(n) = (π/10)n, modulo 2π plotted in the range (−π, π].
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Classification of Discrete-Time Signals
Energy signals and power signals
Energy E of a signal x(n)
E ≡
∞
n=−∞
|x(n)|2
If E is finite, x(n) is called an energy signal
Many signals with infinite energy have a finite average power
Average power of x(n)
P = lim
N→∞
1
2N + 1
N
n=−N
|x(n)|2
If P is finite (and nonzero), x(n) is called a power signal
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Classification of Discrete-Time Signals
Example
Power and energy of unit step sequence
P = lim
N→∞
1
2N + 1
N
n=0
u2
(n) = lim
N→∞
N + 1
2N + 1
= lim
N→∞
1 + 1/N
2 + 1/N
=
1
2
It is a power signal (its energy is infinite)
Example
Power and energy of complex exponential sequence x(n) = Aejω0n
P = lim
N→∞
1
2N + 1
N
n=−N
A2
= lim
N→∞
(2N + 1)A2
2N + 1
= A2
It is a power signal
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Classification of Discrete-Time Signals
Periodic signals and aperiodic signals
x(n) is periodic with period N (N > 0) if and only if
x(n + N) = x(n) for all n
Smallest value of N is called fundamental period
If there is no value of N that satisfies above equation, signal is called
aperiodic
Remember x(n) = A sin 2πf0n is periodic if f0 = k
N = rational number
Energy of a periodic signal over a single period is finite if it takes on
finite values
It is infinite for −∞ ≤ n ≤ ∞
Average power of a periodic signal is finite
Equal to average power over a single period
If x(n) is periodic with fundamental period N and takes on finite values
P =
1
N
N−1
n=0
|x(n)|2
Periodic signals are power signals
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Classification of Discrete-Time Signals
Symmetric (even) and antisymmetric (odd) signals
Real-valued signal x(n) is symmetric (even) if
x(−n) = x(n)
x(n) is antisymmetric (odd) if
x(−n) = −x(n)
x(0) = 0
Any arbitrary signal can be expressed as sum of one even and one odd
signal components
xe(n) =
1
2
[x(n) + x(−n)]
xo(n) =
1
2
[x(n) − x(−n)]
x(n) = xe(n) + xo(n)
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Classification of Discrete-Time Signals
Figure 7: Example of even (a) and odd (b) signals.
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Simple Manipulations of Discrete-Time Signals
Transformation of independent variable (time)
x(n) is shifted in time by replacing n by n − k
If k > 0 −→ delay of signal by k units of time
If k < 0 −→ advance of signal by |k| units in time
x(n) is folded or reflected about time origin n = 0 by replacing n by −n
Operations of folding (FD) and time delaying (TD) (or advancing) a
signal are not commutative
TDk [x(n)] = x(n − k), k > 0
FD[x(n)] = x(−n)
TDk {FD[x(n)]} = TDk [x(−n)] = x(−n + k)
FD{TDk [x(n)]} = FD[x(n − k)] = x(−n − k)
x(n) is time scaled or down-sampled by replacing n by µn where µ is
an integer
If y(n) = x(2n)
we know x(n) = xa(nT)
y(n) = x(2n) = xa(2Tn)
Hence this time-scaling operation is equivalent to changing sampling
rate from 1/T to 1/2T −→ a down-sampling operation
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Simple Manipulations of Discrete-Time Signals
Figure 8: Graphical representation of a signal, and its delayed and advanced
versions. 16 / 49
Simple Manipulations of Discrete-Time Signals
Figure 9: Graphical illustration of the folding and shifting operations.
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Simple Manipulations of Discrete-Time Signals
Figure 10: Graphical illustration of down-sampling operation.
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Simple Manipulations of Discrete-Time Signals
Amplitude modifications
Amplitude scaling by a constant A
y(n) = Ax(n), −∞ < n < ∞
Sum of two signals
y(n) = x1(n) + x2(n), −∞ < n < ∞
Product of two signals
y(n) = x1(n)x2(n), −∞ < n < ∞
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Discrete-Time Systems
Discrete-time system
A device or algorithm that operates on a discrete-time signal called
input or excitation, according to some well-defined rule, to produce
another discrete-time signal called output or response of system
Input signal x(n) is transformed by system into output signal y(n)
y(n) ≡ τ[x(n)]
Figure 11: Block diagram representation of a discrete-time system.
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Input-Output Description of Systems
Input-output description of a system
Consists of a mathematical expression or a rule defining relation between
input and output signals
The only way to interact with system is by using its input and output
terminals
System is assumed to be a black box
Exact internal structure of system is either unknown or ignored
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Input-Output Description of Systems
Example
Response of following systems to the input signal
x(n) =
|n|, −3 ≤ n ≤ 3
0, otherwise
x(n) = {. . . , 0, 3, 2, 1, 0
↑
, 1, 2, 3, 0, . . .}
1 y(n) = x(n) (identity system)
y(n) = x(n) = {. . . , 0, 3, 2, 1, 0
↑
, 1, 2, 3, 0, . . .}
2 y(n) = x(n − 1) (unit delay system)
y(n) = {. . . , 0, 3, 2, 1
↑
, 0, 1, 2, 3, 0, . . .}
3 y(n) = x(n + 1) (unit advance system)
y(n) = {. . . , 0, 3, 2, 1, 0, 1
↑
, 2, 3, 0, . . .}
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Input-Output Description of Systems
Example (continued)
4 y(n) = 1
3[x(n + 1) + x(n) + x(n − 1)] (moving average filter)
y(n) = {. . . , 0, 1, 5
3, 2, 1, 2
3
↑
, 1, 2, 5
3, 1, 0, . . .}
E.g., y(0) = 1
3[x(−1) + x(0) + x(1)] = 1
3[1 + 0 + 1] = 2
3
5 y(n) = median{x(n + 1), x(n), x(n − 1)} (median filter)
y(n) = {. . . , 0, 2, 2, 1, 1
↑
, 1, 2, 2, 0, . . .}
6 y(n) = n
k=−∞ x(k) = x(n) + x(n − 1) + x(n − 2) + · · · (accumulator)
y(n) = {. . . , 0, 3, 5, 6, 6
↑
, 7, 9, 12, . . .}
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Input-Output Description of Systems
For some systems, output at n = n0 depends not only on input at
n = n0, but on input values before and after n = n0
E.g., for accumulator
y(n) =
n
k=−∞
x(k) =
n−1
k=−∞
x(k) + x(n) = y(n − 1) + x(n)
Given input signal x(n) for n ≥ n0, output y(n) for n ≥ n0
y(n0) = y(n0 − 1) + x(n0)
y(n0 + 1) = y(n0) + x(n0 + 1)
and so on
The additional information required to determine y(n) for n ≥ n0 is
initial condition y(n0 − 1)
With no excitation prior to n0, initial condition is y(n0 − 1) = 0
System is initially relaxed
Every system is relaxed at n = −∞
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Input-Output Description of Systems
Example
Following accumulator is excited by sequence x(n) = nu(n)
y(n) =
n
k=−∞
x(k)
Output of system
y(n) =
−1
k=−∞
x(k)+
n
k=0
x(k) = y(−1)+
n
k=0
x(k) = y(−1)+
n(n + 1)
2
If system is initially relaxed −→ y(−1) = 0
y(n) = n(n+1)
2 , n ≥ 0
If initial condition is y(−1) = 1
y(n) = 1 + n(n+1)
2 = n2+n+2
2 , n ≥ 0
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Block Diagram Representation of Discrete-Time Systems
Symbols used to denote different basic building blocks
An adder
This operation is memoryless (not necessary to store sequences)
A constant multiplier (memoryless operation)
A signal multiplier (memoryless operation)
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Block Diagram Representation of Discrete-Time Systems
Symbols used to denote different basic building blocks
A unit delay element (requires memory)
A unit advance element (requires memory)
Example
Using basic building blocks, sketch block diagram of
y(n) = 1
4y(n − 1) + 1
2x(n) + 1
2x(n − 1)
Shown in Fig. 12 (a)
A simple rearrangement
y(n) = 1
4y(n − 1) + 1
2[x(n) + x(n − 1)]
Shown in Fig. 12 (b)
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Block Diagram Representation of Discrete-Time Systems
Example (continued)
Figure 12: Block diagram realizations of the system
y(n) = 0.25y(n − 1) + 0.5x(n) + 0.5x(n − 1).
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Classification of Discrete-Time Systems: Static vs Dynamic
Static or memoryless system
Output at any instant n depends at most on input sample at same time,
but not on past or future samples of input
Dynamic
A system which is not static
Has memory
If output at time n is completely determined by input samples from
n − N to n (N ≥ 0), system is said to have memory of duration N
N = 0 −→ system is static
0 < N < ∞ −→ system has finite memory
N = ∞ −→ system has infinite memory
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Classification of Discrete-Time Systems: Static vs Dynamic
Example
Following systems are static
y(n) = ax(n)
y(n) = nx(n) + bx3(n)
Following systems are dynamic
y(n) = x(n) + 3x(n − 1)
This system has finite memory
y(n) = n
k=0 x(n − k)
This system has finite memory
y(n) = ∞
k=0 x(n − k)
This system has infinite memory
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Classification of D-T Systems:Time-Invariant,Time-Variant
A relaxed system τ is time invariant or shift invariant if and only if
x(n)
τ
→ y(n)
implies that
x(n − k)
τ
→ y(n − k)
for every input signal x(n) and every time shift k
To determine if any given system is time invariant
1 Excite system with an arbitrary sequence x(n), which produces y(n)
2 Delay input sequence by some amount k and recompute output
y(n, k) = τ[x(n − k)]
3 If y(n, k) = y(n − k), for all possible k, system is time invariant. If not,
even for one k, system is time variant
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Classification of D-T Systems:Time-Invariant,Time-Variant
Example
Is this system time invariant or time variant?

1
z
)1()()(  nxnxny)(nx

Input-output equation of system
y(n) = τ[x(n)] = x(n) − x(n − 1)
Delaying input by k units, it is clear from block diagram that
y(n, k) = x(n − k) − x(n − k − 1)
On the other hand, delaying y(n) by k units
y(n − k) = x(n − k) − x(n − k − 1)
Since y(n, k) = y(n − k), system is time invariant
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Classification of D-T Systems:Time-Invariant,Time-Variant
Example
Is this system time invariant or time variant?
)(nx

)()( nnxny 
n
Input-output equation of system
y(n) = τ[x(n)] = nx(n)
Response of this system to x(n − k) is
y(n, k) = nx(n − k)
If we delay y(n) by k units
y(n − k) = (n − k)x(n − k) = nx(n − k) − kx(n − k)
Since y(n, k) = y(n − k), system is time variant
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Classification of D-T Systems:Time-Invariant,Time-Variant
Example
Is this system time invariant or time variant?
)(nx )()( nxny 

Input-output equation of system
y(n) = τ[x(n)] = x(−n)
Response of this system to x(n − k) is
y(n, k) = τ[x(n − k)] = x(−n − k)
If we delay y(n) by k units
y(n − k) = x(−n + k)
Since y(n, k) = y(n − k), system is time variant
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Classification of D-T Systems:Time-Invariant,Time-Variant
Example
Is this system time invariant or time variant?
)(nx

nnxny 0cos)()( 
n0cos
Input-output equation of system
y(n) = x(n) cos ω0n
Response of this system to x(n − k) is
y(n, k) = x(n − k) cos ω0n
If we delay y(n) by k units
y(n − k) = x(n − k) cos ω0(n − k)
Since y(n, k) = y(n − k), system is time variant
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Classification of D-Time Systems: Linear vs Nonlinear
A system is linear if and only if
τ[a1x1(n) + a2x2(n)] = a1τ[x1(n)] + a2τ[x2(n)]
for any arbitrary input sequences x1(n) and x2(n), and any arbitrary
constants a1 and a2
A linear system satisfies superposition principle
This principle requires that response of system to a weighted sum of
signals be equal to the corresponding weighted sum of responses of
system to each of individual input signals
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Classification of D-Time Systems: Linear vs Nonlinear
Figure 13: Graphical representation of the superposition principle. τ is linear if
and only if y(n) = y (n).
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Classification of D-Time Systems: Linear vs Nonlinear
Linearity condition (superposition principle)
τ[a1x1(n) + a2x2(n)] = a1τ[x1(n)] + a2τ[x2(n)]
Suppose a2 = 0
τ[a1x1(n)] = a1τ[x1(n)] = a1y1(n)
This is multiplicative or scaling property of a linear system
If a1 = 0, then y(n) = 0 −→ a relaxed, linear system with zero input
produces a zero output
Suppose a1 = a2 = 1
τ[x1(n) + x2(n)] = τ[x1(n)] + τ[x2(n)] = y1(n) + y2(n)
This is additivity property of a linear system
Extension of linearity condition
x(n) =
M−1
k=1
ak xk (n)
τ
−→ y(n) =
M−1
k=1
ak yk (n)
where yk (n) = τ[xk (n)], k = 1, 2, . . . , M − 1
If a relaxed system does not satisfy superposition principle, it is
nonlinear
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Classification of D-Time Systems: Linear vs Nonlinear
Example
Determine if y(n) = nx(n) is linear or nonlinear
For two inputs x1(n) and x2(n), outputs are
y1(n) = nx1(n)
y2(n) = nx2(n)
A linear combination of two input sequences results in output
y3(n) = τ[a1x1(n) + a2x2(n)] = n[a1x1(n) + a2x2(n)] =
a1nx1(n) + a2nx2(n)
A linear combination of two output sequences results in output
a1y1(n) + a2y2(n) = a1nx1(n) + a2nx2(n)
Since right-hand sides of two above equations are identical, system is
linear
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Classification of D-Time Systems: Linear vs Nonlinear
Example
Determine if y(n) = x(n2) is linear or nonlinear
Response of system to two separate inputs x1(n) and x2(n)
y1(n) = x1(n2)
y2(n) = x2(n2)
Output of system to a linear combination of x1(n) and x2(n)
y3(n) = τ[a1x1(n) + a2x2(n)] = a1x1(n2) + a2x2(n2)
A linear combination of two output sequences
a1y1(n) + a2y2(n) = a1x1(n2) + a2x2(n2)
Since right-hand sides of two above equations are identical, system is
linear
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Classification of D-Time Systems: Linear vs Nonlinear
Example
Determine if y(n) = x2(n) is linear or nonlinear
Response of system to two separate inputs
y1(n) = x2
1 (n)
y2(n) = x2
2 (n)
Response of system to a linear combination of these two inputs
y3(n) = τ[a1x1(n) + a2x2(n)] = [a1x1(n) + a2x2(n)]2 =
a2
1x2
1 (n) + 2a1a2x1(n)x2(n) + a2
2x2
2 (n)
If system is linear, it will produce a linear combination of two outputs
a1y1(n) + a2y2(n) = a1x2
1 (n) + a2x2
2 (n)
Since right-hand sides of two above equations are not identical, system
is nonlinear
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Classification of D-Time Systems: Linear vs Nonlinear
Example
Determine if y(n) = Ax(n) + B is linear or nonlinear
For two inputs x1(n) and x2(n), outputs are
y1(n) = Ax1(n) + B
y2(n) = Ax2(n) + B
A linear combination of x1(n) and x2(n) results in output
y3(n) = τ[a1x1(n) + a2x2(n)] = A[a1x1(n) + a2x2(n)] + B =
a1Ax1(n) + a2Ax2(n) + B
If system were linear, its output would be
a1y1(n) + a2y2(n) = a1Ax1(n) + a1B + a2Ax2(n) + a2B
The two results are different and system fails to satisfy linearity test.
Reason is not that system is nonlinear but with B = 0 system is not
relaxed.
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Classification of D-Time Systems: Linear vs Nonlinear
Example
Determine if y(n) = ex(n) is linear or nonlinear
This system is relaxed
If x(n) = 0 −→ y(n) = 1
Hence system is nonlinear
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Classification of D-Time Systems: Causal vs Noncausal
A system is causal if its output at any time depends only on present
and past inputs but not on future inputs
y(n) = F[x(n), x(n − 1), x(n − 2), . . .]
If a system does not satisfy this definition, it is noncausal
Example
These systems are causal
y(n) = x(n) − x(n − 1)
y(n) = n
k=−∞ x(k)
y(n) = ax(n)
These systems are noncausal
y(n) = x(n) + 3x(n + 4)
y(n) = x(n2)
y(n) = x(2n)
y(n) = x(−n)
e.g., n=−1
−−−−−−−→ y(−1) = x(1)
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Classification of D-Time Systems: Stable vs Unstable
A relaxed system is bounded input-bounded output (BIBO) stable if
and only if every bounded input produces a bounded output
x(n) and y(n) are bounded if there exist some finite numbers, Mx and
My , such that for all n
|x(n)| ≤ Mx < ∞, |y(n)| ≤ My < ∞
If for bounded x(n), output is unbounded (infinite), system is unstable
Example
Consider nonlinear system
y(n) = y2(n − 1) + x(n)
We select bounded input
x(n) = Cδ(n)
where C is a constant. Assume y(−1) = 0. Output sequence is
y(0) = C, y(1) = C2, y(2) = C4, . . . , y(n) = C2n
Output is unbounded when 1 < |C| < ∞
System is BIBO unstable
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Interconnection of Discrete-Time Systems
Systems can be interconnected in two ways to form larger systems
Cascade (series)
Parallel
)(nx )(ny
1
)(1 ny
2
c
)(nx
1
)(1 ny
p

2
)(2 ny
)(3 ny
Figure 14: Cascade and parallel interconnections of systems.
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Interconnection of Discrete-Time Systems
In cascade interconnection
Output of first system is
y1(n) = τ1[x(n)]
Output of second system
y(n) = τ2[y1(n)] = τ2[τ1[x(n)]]
Combining systems τ1 and τ2 into a single system τc
τc ≡ τ2τ1 −→ y(n) = τc [x(n)]
For arbitrary systems τ1 and τ2
τ2τ1 = τ1τ2
If systems τ1 and τ2 are linear and time invariant, then
1 τc is time invariant
x(n − k)
τ1
−→ y1(n − k)
and
y1(n − k)
τ2
−→ y(n − k)
thus
x(n − k)
τc =τ2τ1
−−−−−→ y(n − k)
2 τ2τ1 = τ1τ2
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Interconnection of Discrete-Time Systems
Output of parallel interconnection is
y3(n) = y1(n)+y2(n) = τ1[x(n)]+τ2[x(n)] = (τ1 +τ2)[x(n)] = τp[x(n)]
where τp = τ1 + τ2
Parallel and cascade interconnections can be used to construct larger,
more complex systems
Conversely, a larger system can be broken down into smaller subsystems
for purposes of analysis and implementation
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References
John G. Proakis, Dimitris G. Manolakis, Digital Signal
Processing: Principles, Algorithms, and Applications, Prentice
Hall, 2006.
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