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Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila

Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila. Some basic system properties Invertibility and inverse systems. Invertibility systems: if distinct inputs leads to distinct outputs .

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Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila

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  1. Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila S&S Lecture # 03 Mariam Shafqat

  2. Some basic system propertiesInvertibility and inverse systems • Invertibility systems: if distinct inputs leads to distinct outputs. • For discrete time case, if input is invertible then an inverse system exists that when cascaded with original system yield an output equal to original input. • For example • y(t)= 2x(t) • w(t)=1/2y(t) S&S Lecture # 03 Mariam Shafqat

  3. Example • W[n]= y[n]-y[n-1]// inverse system for accumulator • Examples of non-invertible systems are y[n]=0 • That is, the system that produces the zero output sequence for any input sequence, and Y[n]=x2(t) • In which case we cannot determine the sign of the input from the knowledge of output. S&S Lecture # 03 Mariam Shafqat

  4. Why use encoder? • Desire to secure the original message for secure or private communication • Provide redundancy in the signal so that error that occurred in the signal can be detected or corrected. • Lossless coding: the input to the encoder must be exactly recoverable from the output i.e. encoder must be exactly invertible. S&S Lecture # 03 Mariam Shafqat

  5. Causality • Def: the output at any time depends upon the input at only the present and past values. • Also called non-anticipative. As output does not anticipate future values of the input. • E.g y[n]=x[n-1] • Motion of an automobile • All memoryless values are casual, since output responds only to the current values of the input. S&S Lecture # 03 Mariam Shafqat

  6. Non-causal systems • Depends upon future values. Might also depends upon the present and past values. • Y[n]= x[n]- x[n+1] S&S Lecture # 03 Mariam Shafqat

  7. Stability • It is the one in which small input leads to responses that do not diverge. • Stability of stable systems generally results from the presence of mechanism that dissipate energy. • y(t)=kx(t) //unstable system S&S Lecture # 03 Mariam Shafqat

  8. Stability Mechanism • Suppose that input x[n] is bounded by some number say B, for all values of n. • Then the largest possible magnitude for y[n] is also bounded by B because y[n] is the average of the finite set of values of the input. • Therefore y[n] is bounded and is thus stable. S&S Lecture # 03 Mariam Shafqat

  9. Time invariance • Behavior and characteristics of the system are fixed over time. • Specifically, a signal is said to be time invariant, if a time shift in the input signal causes an identical time shift in the output signal as well. • For example y[n] is the output of a discrete time time-invariant system, x[n] is the input. Then y[n-n0] is the output of x[n-n0] S&S Lecture # 03 Mariam Shafqat

  10. Time Invariance • Similarly for the continuous time case • If y(t) is the output corresponding to the input x(t), a time-invariant system will have y(t-t0) for the input x(t-t0). S&S Lecture # 03 Mariam Shafqat

  11. Linearity • Is a system that obeys the property of superposition. • If the input consists of weighted sum of several signals, then the output is the superposition ---- that is the weighted sum--- of the responses of the system to each of those signals. • Let y1(t) be the responses of a continuous time systems to an input x1(t), and let y2(t) be the input to the input x2(t.) then the system is linear if: S&S Lecture # 03 Mariam Shafqat

  12. Linearity • The response to x1(t)+x2(t) is y1(t)+y2(t). • The response to ax1(t) is ay1(t), where a is any complex constant. • The first property is called additive property • Second property is called scaling or homogeneity property. • Superposition=additive + homogeneity. • Same discussion holds for discrete time signals. S&S Lecture # 03 Mariam Shafqat

  13. Linearity • Continuous time signals: ax1(t)+bx2(t)=ay1(t)+by2(t) • Discrete Time signals: ax1[n]+bx2[n]=ay1[n]+by2[n] • Zero-input response ( an input that is zero for all time results in an output which is zero for all time.) • 0.x[n]=0 • 0.x(t)=0 S&S Lecture # 03 Mariam Shafqat

  14. Linearity Examples • Linear Systems: • Y(t)=tX(t) • Y[n]=2x[n] • Non—linear systems: • Y[n]=2x[n]+3 • Y(t)=Re{x(t)} S&S Lecture # 03 Mariam Shafqat

  15. Example 1.17 • Y(t)=t x(t) • X1(t) y1(t)=tx1(t) • X2(t) y2(t)=tx2(t) • X3(t)=ax1(t)+bx2(t) • Where a and b are arbitrary scalars • Y3(t)=tx3(t) • T(ax1(t)+bx2(t)) • aTx1(t)+bTx2(t) • ay1(t)+by2(t) S&S Lecture # 03 Mariam Shafqat

  16. Example 1.18 • Y(t)=x2(t) • X1(t)y1(t)=x12(t) • X2(t)y2(t)=x22(t) • And x3(t)y3(t)=x32(t) • And x3(t)y3(t)=x32(t) • (ax1(t)+bx2(t))2 • a2x12(t)+ b2x22(t)+2abx1(t)x2(t) • a2y1(t)+b2y2(t)+2abx1(t)x2(t) S&S Lecture # 03 Mariam Shafqat

  17. Example 1.20 • Y[n]=2x[n]+3 S&S Lecture # 03 Mariam Shafqat

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