235Concept Check

3.5 Series and Parallel Interconnection of Systems

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What is the impulse response of the series of two systems?

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How will you find the impulse response of the parallel configuration?

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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.

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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.

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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 are

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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.

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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

1. The system is linear if

(a) it is homogeneous

(b) bit is additive

(d) it is additive and homogeneous

2. The system is causal when the current output sample depends on

(a) current input sample

(b) current or next and past input samples

3. The range of values of “a” for which the system with impulse response h(n) = an_{u(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

(d) the system may be homogeneous

5. The system of human being is

(a) non-causal (b) non-linear

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6. The system is said to be memoryless if

(a) Only on a current input sample

(b) current or next and past input samples

7. The following system is invertible

(a) different transforms

(b) all systems

8. The bench mark system can be designed using

(a) causal systems (b) invertible systems

9. The following systems are prone to noise

(a) non-causal systems (b) causal systems

10. The system is BIBO stable if

(a) the output is bonded for every bounded input

(b) the output is always bounded

(d) the output decreases

11. The system is called as LTI if (a) the system is linear

(b) the system is linear and time invariant (c) the system is time invariant

(d) the system is additive 12. The system given by y(t) = x(t) + 5

(a) is memoryless

(b) is with memory

(d) is stable and with memory

13. The system given by y[n] = x[3 – n] is

(a) causal (b) anti causal

14. The system given by y(t) = x2_{(t) is}

(a) invertible (b) non-invertible

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15. The system given by y[n] = (n – 1)x[n] is

(a) time invariant (b) time dependant

16. The system given by y[n] = x[n – 1]sin(nω) is

(a) time invariant (b) LTI

17. Series interconnection of system results in (a) addition of the impulse responses

(b) convolution of impulse responses

18. Parallel interconnection of systems results in (a) addition of the impulse responses

(b) convolution 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.

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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] (b) y[n] = nx[n] (c) y[n] = x[1 – n] (d) y[n] = x[n]sin(wn) (e) y(t) = x(t) + x(t + 1) (f) y(t) = t2_x(t) (g) y(t) = x(4 – t) (h) y(t) = x(t)sin(t)

3.6 Find if the following systems are linear. (a) y[n] = (n + 1)x[n] (b) y[n] = x[n2_] (c) y[n] = x3_[n] (d) y[n] = 2x[n] + 3 (e) y(t) =(t + 2)x(t) (f) y(t) = x3_(t) (g) y(t) = 3x(t) + 1 (h) y(t) = sin(t)x(t)

3.7 Find if the following systems are causal. (a) y[n] = 5x[n] (b) + =−∞ =

∑

1 [ ] n ( ) k y n x k

Signals and Systems 240 (c) y[n] = x[3 – n] (d) y[n] = x[3n] (e) y(t) = x(t2₎ (f) y(t) = x(5 – t) (g) y(t) = x(2t – 2) (h) y(t) = x(–2t)

3.8 Find if the following systems are memoryless (a) y(t) = e–2_x(t) (b) y(t) = cos(x(t)) (c) y[n] = 5x[n] + 2x[n]u[n] (d) τ τ −∞ =

_∫

/3 ( ) t ( ) y t x d (e) y(t) = x(7 – 2t) (f) y(t) = x(t/5)

3.9 Find if the following systems are stable. (a) y(t) = cos(x(t))

(b) y[n] = log₁₀ (|x[n]|) (c) y[n] = cos(2p x[n]) + x[n] (d) y t( )=_dtd[e x t−t ( )] (e) y(t) = x(t/3) (f) =−∞ =

∑

+ [ ] n [ 3] m y n x m (g) ∞ δ =−∞ =

∑

− [ ] [ ] [ 5 ] m y n x n n m

3.10 Find if the following systems are invertible. (a) −∞ =

∑

+ [ ] n [ 3] m y n x m (b) y n x n[ ]= [ 1] 4− + (c) y(t) = x3_(t) (d) y t( )=x t( / 9)

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(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)

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(d)

(e)

3.13 Find the possible interconnection for the following equation of the overall impulse response of the system.

(a) h_overall[ ] {[ [ ]n = h n h n₁ + ₂[ ]]*[ [ ]h n h n₃ + ₁[ ]]}* [ ]h n₁ (b) h_overall[ ] {[ [ ]* [ ]] [ [ ]* [ ]]}*[ [ ]n = h n h n₁ ₂ + h n h n₃ ₁ h n h n₁ + ₂[ ]] (c) _h_overall_{( ) { ( ) [ ( )* ( )]}* ( )}_t = _{h t}₁ + _{h t h t}₂ ₃ _{h t}₃

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

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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)

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3.11 (2)

3.11 (3)

C T and DT Systems 245 3.12 (a) h_overall( ) {[ ( )t = h t h t₁ + ₂( )]* ( )h t₃ (b) h_overall( ) {[ ( )* ( )] [ ( )* ( )]}* ( )t = h t h t₁ ₂ + h t h t₂ ₃ h t₃ (c) h_overall( ) { ( ) [ ( )* ( )]}* ( )* ( )t = h t₁ + h t h t₂ ₃ h t h t₃ ₂ (d) h_overall_{[ ] {[ [ ]] [ [ ]* [ ]]}* [ ]}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)

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