= + +
1 1
1 / 1
2 3
H S S and = + +
1 1
[ ] 1 / 1 [ ]
2 3
y n S S x n (3.123)
Note that we have to take the term containing y(n – 1) on the left-hand side to combine it with y(n). The representation is clear from Fig. 3.7.
Fig. 3.7 Block schematic for Example 3.50
Concept Check
•
What is a moving average system?•
How will you divide the system operator into number of operators such as time sifting operator?•
What is the advantage of using interconnection of operations?3.5 Series and Parallel Interconnection of Systems
Let us understand the significance of series and parallel interconnections of the systems. We need to understand what will happen to the impulse response of the interconnected system. These interconnections have significance for LTI systems.
CT and DT Systems 225 3.5.1 Series interconnection of systems
Consider the series connection of two linear and shift invariant systems, i.e., LTI CT systems, as shown in Fig. 3.8. Let the impulse responses of the two systems be specified as h1(t) and h2(t), respectively. The series connection is also termed as cascaded configuration. The output of the system is y(t). Let z(t) represent the output of the first system with impulse response h1(t). The output z(t) can be written as
τ τ τ
Here, the symbol * denotes the convolution operation. The meaning of convolution and the procedure to implement convolution in CT and DT domain will be discussed in detail in chapter 4.
The output of first system z(t) is applied as input to the second system with impulse response h2(t). The output of the second system y(t) will be given as
τ τ τ
Signals and Systems 226
We can conclude that when the two systems are cascaded or connected in series, the impulse response of the cascaded system is the convolution of impulse responses of the two systems.
Fig. 3.8 Series interconnection of CT systems
In case of DT systems, same equations will hold good. When the two systems with impulse responses h1[n] and h2[n] are cascaded or connected in series, the impulse response of the cascaded system is the convolution of impulse responses of the two systems given by h1[n]*h2[n].
Consider the series connection of two linear and shift invariant systems, i.e., LTI DT systems, as shown in Fig. 3.9. Let the impulse responses of the two systems be specified as h1[n] and h2[n], respectively. The series connection is also termed as cascaded configuration. The output of the system is y[n]. Let z[n] represent the output of the first system with impulse response h1[n]. The output z[n] can be written as
=−∞
The output of first system z[n] is applied as input to the second system with impulse response h2[n]. The output of the second system y[n] will be given as
=−∞
CT and DT Systems
We can conclude that when the two systems are cascaded or connected in series, the impulse response of the cascaded system is the convolution of impulse responses of the two systems.
Fig. 3.9 Series interconnection of DT systems
3.5.2 Parallel interconnection of systems
Consider the parallel connection of two linear and shift invariant systems, i.e., LTI systems, as shown in Fig. 3.10. Let the impulse responses of the two systems be specified as h1(t) and h2(t), respectively.
Fig. 3.10 Parallel interconnection of CT systems
The output of the system is y(t). Let z(t) represent the output of the first system with impulse response h1(t). Let w(t) represent the output of the second system with impulse response h(t). The output z(t) can be written as
Signals and Systems
The symbol * is used to represent the convolution operation.
The output w(t) can be written as w(t) = x(t) *h2(t)
The output of the parallel interconnection y(t) can be written as
η η η η η η
Let the impulse response of the parallel configuration be written as h(t) given by h(t) = h1 (t) + h2(t)
We can conclude that when the two systems are connected in parallel, the impulse response of the parallel configuration system is the addition of impulse responses of the two systems.
In case of DT systems, same equations will hold good. When the two systems with impulse responses h1[n] and h2[n] are connected in parallel, the impulse response of the cascaded system is the addition of impulse responses of the two systems given by h1[n] + h2[n].
Consider the parallel connection of two linear and shift invariant systems, i.e., LTI systems, as shown in Fig. 3.11. Let the impulse responses of the two systems be specified as h1[n] and h2[n], respectively.
CT and DT Systems 229 Fig. 3.11 Parallel interconnection of DT systems
The output of the system is y[n]. Let z[n] represent the output of the first system with impulse response h1[n]. Let w[n] represent the output of the second system with impulse response h2[n]. The output z[n] can be written as
z[n] = x[n]*h1[n]
The symbol * is used to represent the convolution operation.
The output w[n] of the second system can be written as w[n] = x[n]*h2[n]
The output of the parallel interconnection y[n] can be written as
=−∞ =−∞
Let the impulse response of the parallel configuration be written as h[n] given by
= 1 + 2 [ ] [ ] [ ]
h n h n h n (3.135)
Signals and Systems
We can conclude that when the two systems are connected in parallel, the impulse response of the parallel configuration system is the addition of impulse responses of the two systems.
Let us go through some numerical problems to illustrate the concepts.
Example 3.51
Consider the configuration shown in Fig. 3.12 with impulse responses given by h1(t), h2(t), h3(t) for the systems, as shown. Find the impulse response of the overall configuration.
Fig. 3.12 Configuration of three systems for Example 3.51 Solution
Firstly we will find the impulse response of the parallel interconnection of two systems with impulse response h1(t) and h2(t). The impulse response of the parallel configuration is h(t) = h1(t) + h2(t). The third system is connected in series with the parallel configuration. Hence, the impulse response of the series configuration will be the convolution of the two impulse responses. We find that the distributive law holds good.
= 1 + 2 3 = 1 3 + 2 3
'( ) [ ( ) ( )]* ( ) ( )* ( ) ( )* ( )
h t h t h t h t h t h t h t h t (3.137)
Example 3.52
Consider the configuration shown in Fig. 3.13 with impulse responses given by h1(t), h2(t), h3(t) for the systems, as shown in the figure below. Find the impulse response of the overall configuration.
CT and DT Systems 231 Fig. 3.13 Configuration of four systems for Example 3.52
Solution
Let us find the impulse response of the series system h(t) giving output as z(t)
= 1 3
( ) ( )* ( )
h t h t h t . (3.138)
This series configuration is in parallel with h2(t). The impulse response of the parallel configuration is
= + 2 = 1 3 + 2
'( ) [ ( ) ( )] [ ( )* ( )] ( )
h t h t h t h t h t h t (3.139)
The system with impulse response h3(t) is connected in series with this parallel configuration. The impulse response of the overall system is then given by
= + 2 3 = 1 3 + 2 3
''( ) [ ( ) ( )]* ( ) {[ ( )* ( )] ( )}* ( )
h t h t h t h t h t h t h t h t (3.140)
Example 3.53
Consider the configuration shown in Fig. 3.14 with impulse responses given by h1(t), h2(t), h3(t) for the systems, as shown in the figure below. Find the impulse response of the overall configuration.
Fig. 3.14 Configuration of four systems for Example 3.53
Signals and Systems 232
Solution
Let us find the impulse response of the parallel system h(t) giving output as z(t)
= 1 + 2 ( ) ( ) ( )
h t h t h t . (3.141)
This parallel configuration is in series with h3(t). The impulse response of the series configuration is
= 1 + 2 3
'( ) [ ( ) ( )]* ( )
h t h t h t h t (3.142)
The system with impulse response h1(t) is connected in series with this series configuration. The impulse response of the overall system is then given by
= 1 + 2 3 1
''( ) [ ( ) ( )]* ( )* ( )
h t h t h t h t h t (3.143)
Example 3.54
Consider the overall impulse response of the system given by
= 1 2 + 3 1
''( ) [ ( )* ( )] [ ( )* ( )]
h t h t h t h t h t for the three systems with impulse responses h1(t), h2(t), h3(t). Draw the configuration.
Solution
The overall impulse response for the configuration indicates that the series combination of h1(t) and h2(t) is connected in parallel with series combination of h3(t) and h1(t). It can be drawn as shown in Fig. 3.15.
Fig. 3.15 Configuration of four systems for Example 3.54 We will now solve some examples for DT systems.
Example 3.55
Consider the configuration shown in Fig. 3.16 with impulse responses given by h1[t], h2[t] and h3[t] in Fig. 3.16 for the systems, as shown in the figure below. Find the impulse response of the overall configuration.
CT and DT Systems 233 Fig. 3.16 Configuration of five systems for Example 3.55
Solution
Let us find the impulse response of the series system h[n] giving output as z[n]
= 1 3
[ ] [ ]* [ ]
h n h n h n. (3.144)
Let us find the impulse response of the series system h¢[n] giving output as w[n]
= 2 3
'[ ] [ ]* [ ]
h n h n h n. (3.145)
These series configurations are in parallel. The impulse response of the parallel configuration h’’[n] is
= + = 1 3 + 2 3
''[ ] [ [ ] '( )] [ [ ]* [ ]] [ [ ]* [ ]]
h n h n h t h n h n h n h n (3.146)
The system with impulse response h3[n] is connected in series with this parallel configuration. The impulse response of the overall system hoverall is then given by
= + = +
overall 3 1 3 2 3 3
[ ] [ [ ] '[ ]]* [ ] {[ [ ]* [ ]] [ [ ]* [ ]]}* ( )
h n h n h n h n h n h n h n h n h t (3.147)
Example 3.56
Consider the configuration shown in Fig. 3.17 with impulse responses given by h1[n], h2[n], h3[n] for the systems, as shown in the figure below. Find the impulse response of the overall configuration.
Solution
Let us find the impulse response of the parallel system h[n] giving output as z[n]
= 1 + 2 [ ] [ ] [ ]
h n h n h n . (3.148)
Signals and Systems 234
Fig. 3.17 Configuration of three systems for Example 3.56
This parallel configuration is in series with h3[n]. The impulse response of the series configuration is
= 1 + 2 3
'( ) [ ( ) ( )]* ( )
h t h t h t h t (3.149)
Example 3.57
Consider the overall impulse response of the system given by h¢¢[n] = {[h1[n]*
h2[n]]+[h3[n]*h1[n]]}*h1[n] for the three systems with impulse responses h1[n], h2[n], h3[n]. Draw the configuration.
Solution
The overall impulse response for the configuration indicates that the series combination of h1[n] and h2[n] is connected in parallel with series combination of h3[n] and h1[n]. This parallel configuration is in series with h1[n]. It can be drawn as shown in Fig. 3.18.
Fig. 3.18 Configuration of five systems for Example 3.57 Things to remember
If the two systems are connected in series, their impulse responses get convolved. If the two systems are connected in parallel, their impulse responses get added.
CT and DT Systems 235 Concept Check
•
What is the impulse response of the series of two systems?•
How will you find the impulse response of the parallel configuration?•
Does the rule for series and parallel interconnection hold good for DT systems?Summary
In this chapter, we have described and explained the properties of DT systems.
•
We have discussed the important properties of DT systems, namely, linearity and shift invariance. It was shown that a system is linear when it is homogeneous as well as additive. It was emphasized that if the transfer curve of the system is linear passing through the origin, then it is linear. Linearity has a meaning more than this. If the system obeys the principle of superposition i.e., additivity and homogeneity, then the system is linear. We then defined the property of time/shift invariance. If the system is linear, the input signal can be suitably decomposed into component signals and the corresponding outputs for the component signals one at a time can be calculated by assuming all other inputs equal to zero. The component outputs can be scaled and added to generate the output of the system for the input signal. This is exactly the property of superposition. The system is said to be time/shift invariant if the input to the system is shifted impulse d(n – k), then it results in a shifted impulse response of h(n – k). The linear and shift invariant system is termed as LTI system. If the system is LTI, one can characterize the system in terms of its impulse response. The calculation of the output for any given input in case of LTI system gets greatly simplified due to the principle of superposition.•
We further concentrated on causality and memory property of systems. We defined the property of causality for systems. The system is said to be causal if the present output of the system depends only on current and past input or output at previous instant. Causal systems are practically realizable or implementable. The system of a human being is causal as we always keep on learning from the past inputs and past outputs of the system. The future inputs have no effect on our act at current or present time. The present and past inputs have meaning only for temporal systems where time is an independent variable. In case of spatial domain systems, present and past input has no meaning. Non-causal temporal systems can be implemented if some delay is tolerable. We can generate a bench mark for system performance using non- causal systems. Offline systems can always be implemented as non-causal systems. The system is said to have memory or said to be dynamic if its current output depends on previous, future input or previous and future output signals. The system is said to be memoryless or instantaneous if its current output depends only on current input. Examples of memoryless systems areSignals and Systems 236
•
We further discussed invertibility and stability. If it is possible to recover the input of the system, then the system is said to be invertible. Invertible system also finds applications in communication field. For error-free transmission an equalizer is used at the input of the receiver that has inverse characteristics as that of the channel. Stability is a notion that describes whether the system will be able to follow the input. A system is said to be unstable if its output is out of control or increases without bound. An arbitrary relaxed system (with zero initial conditions) is said to be bounded input bounded output (BIBO) stable if and only if its output is bounded for every bounded input.•
The system can be described as an interconnection of operations. If the time delay is represented as S block, we can draw the block schematic for the time difference equations. We have discussed series and parallel interconnections of LTI systems and have shown that for series connections, the impulse responses of the individual systems get convolved and for parallel interconnections, the impulse responses of individual systems get added.Multiple Choice Questions
2. The system is causal when the current output sample depends on (a) current input sample
(b) current or next and past input samples
(c) current and/or past input samples and/or past output samples (d) next or past input samples or past output samples
3. The range of values of “a” for which the system with impulse response h(n) = anu(n) is stable is
(a) |a| > 1 (b) |a| < 1
(c) a > 0 (d) a < 0
4. If the transfer graph for a system is linear and passes through origin (a) the system is nonlinear
(b) the system s linear
(c) the system may be additive (d) the system may be homogeneous
5. The system of human being is
(a) non-causal (b) non-linear
(c) causal (d) non-linear and non-causal
CT and DT Systems 237 6. The system is said to be memoryless if
(a) Only on a current input sample (b) current or next and past input samples
(c) current and/or past input samples and/or past output samples (d) next or past input samples or past output samples
7. The following system is invertible (a) different transforms
(b) all systems
(c) if the equation exists to find signal values (d) non-linear system such as squaring device
8. The bench mark system can be designed using
(a) causal systems (b) invertible systems
(c) non-invertible systems (d) non-causal systems 9. The following systems are prone to noise
(a) non-causal systems (b) causal systems (c) non stable systems (d) invertible systems
10. The system is BIBO stable if
(a) the output is bonded for every bounded input
(b) the system is linear and time invariant (c) the system is time invariant
(c) invertible and causal (d) causal and non-invertible 14. The system given by y(t) = x2(t) is
(a) invertible (b) non-invertible
(c) invertible with memory (d) invertible and memoryless
Signals and Systems 17. Series interconnection of system results in
(a) addition of the impulse responses (b) convolution of impulse responses (c) subtraction of impulse responses (d) multiplication of impulse responses
18. Parallel interconnection of systems results in (a) addition of the impulse responses (b) convolution of impulse responses (c) subtraction of impulse responses (d) multiplication of impulse responses
Review Questions
3.1 What is linearity? Define additivity and homogeneity. Is the transfer curve for a linear system always linear? Explain physical significance of linearity.
3.2 What is the time/shift invariance property of systems? Explain physical significance of shift invariance property.
3.3 Explain property of superposition.
3.4 When will you say that the system is memoryless? Give one example of a memoryless system.
3.5 Define causality for a system. Can we design and use a non-causal system?
Is a causal system a requirement for spatial systems?
3.6 Explain the meaning of causality for a system of a human being.
3.7 What is invertibility? Can we use a non-invertible transform for processing a signal?
3.8 Explain the meaning of BIBO stability for a system.
3.9 Explain the physical significance of stability.
3.10 How will you interpret the system as interconnection of operators? Explain using a suitable example.
3.11 Find the impulse response for a series interconnected and parallel interconnected systems. Prove that the impulse response of the series interconnection of two LTI CT systems is a convolution of the two impulse responses.
CT and DT Systems 239 3.12 Find the impulse response for a series interconnected and parallel
interconnected systems. Prove that the impulse response of the series interconnection of two LTI DT systems is a convolution of the two impulse responses.
Problems
3.1 Is the system given by y[n] = x[–n] a linear and shift invariant system?
3.2 Is the system given by y(t) = x(t – 2) a linear and shift invariant system?
3.3 Verify that the systems given by y(n) = x[n]cos(wn) and y[n] = nx[n] are shift variant.
3.4 Check if the systems given by y(t) = (t – 1) x(t) and y(t) = x(t)cos(wt + p /4) are shift invariant?
3.5 Find if the following systems are time invariant.
(a) y[n] = x[n] – x[n – 1]
3.6 Find if the following systems are linear.
(a) y[n] = (n + 1)x[n]
3.7 Find if the following systems are causal.
(a) y[n] = 5x[n]
Signals and Systems
3.8 Find if the following systems are memoryless (a) y(t) = e–2x(t)
3.9 Find if the following systems are stable.
(a) y(t) = cos(x(t))
3.10 Find if the following systems are invertible.
(a)
CT and DT Systems 241 (e) y t( )= x t( )
(f) y[n] = x[2n]
3.11 Represent the following systems in terms of interconnection of operators (a) y(t) = x(t) + x(t – 3) + y(t – 6)
(b) y(t) = x(t – 1) –y(t – 2) –y(t – 3) (c) y[n] = x[n] + y[n – 1] + y[n – 2]
(d) y[n] = x[n – 2] + y[n – 2] – y[n – 4]
3.12 Find the overall impulse response for the interconnection of three systems.
(a)
(b)
(c)
Signals and Systems 242
(d)
(e)
3.13 Find the possible interconnection for the following equation of the overall impulse response of the system.
(a) hoverall[ ] {[ [ ]n = h n h n1 + 2[ ]]*[ [ ]h n h n3 + 1[ ]]}* [ ]h n1 (b) hoverall[ ] {[ [ ]* [ ]] [ [ ]* [ ]]}*[ [ ]n = h n h n1 2 + h n h n3 1 h n h n1 + 2[ ]]
(c) hoverall( ) { ( ) [ ( )* ( )]}* ( )t = h t1 + h t h t2 3 h t3
Answers
Multiple Choice Questions
1 (a) 2 (c) 3 (b) 4 (b) 5 (c) 6 (a) 7 (a) 8 (d) 9 (c) 10 (a)
11 (b) 12 (a) 13 (b) 14 (b) 15 (d)
16 (d) 17 (b) 18 (a)
Problems
3.1 Yes – Linear and shift invariant 3.2 Yes – linear and shift invariant
CT and DT Systems 243 3.3 Yes, systems are time variant
3.4 The systems are time variant 3.5
(a) Yes (b) No (c) Yes (d) No (e) Yes (f) No (g) Yes (h) No
3.6
(a) Yes (b) Yes (c) No (d) Yes (e) Yes (f) No (g) No (h) No
3.7.
(a) Yes (b) No (c) No (d) No (e) No
(f) No (g) No (h) No 3.8
(a) Yes (b) Yes (c) Yes (d) No (e) No (f) No
3.9
(a) Yes (b) Yes (c) Yes (d) Yes (e) Yes (f) No (g) Yes
3.10
(a) No (b) Yes (c) Yes (d) Yes (e) No (f) No
3.11 (1)
Signals and Systems 244
3.11 (2)
3.11 (3)
3.11 (4)
CT and DT Systems 245 3.12 (a) hoverall( ) {[ ( )t = h t h t1 + 2( )]* ( )h t3
(b) hoverall( ) {[ ( )* ( )] [ ( )* ( )]}* ( )t = h t h t1 2 + h t h t2 3 h t3 (c) hoverall( ) { ( ) [ ( )* ( )]}* ( )* ( )t = h t1 + h t h t2 3 h t h t3 2 (d) hoverall[ ] {[ [ ]] [ [ ]* [ ]]}* [ ]n = h n1 + h n h n3 1 h n1 (e) h n''[ ] {[ [ ]* [ ]] [ [ ]* [ ]]}= h n h n1 2 + h n h n3 1 3.13 (a)
(b)
(c)