Signals and Systems - Electrical Engineering

484 C H A P T E R 8: Discrete-Time Signals and Systems The time invariance is shown by letting the input bev[n]=x[n−N],n≥N, and ...

8.3 Discrete-Time Systems 485 Solution For an initial conditiony[−1]=−2 andx[n]=u[n], we get recursively y[0]=0.5y[−1]+x[0]+x[−1 ...

486 C H A P T E R 8: Discrete-Time Signals and Systems For the zero initial condition, it is clear thaty 1 [n]= 2 y[n] when we d ...

8.3 Discrete-Time Systems 487 the first derivative of a continuous-time functionvc(t)as dvc(t) dt ≈ vc(t)−vc(t−Ts) Ts and its se ...

488 C H A P T E R 8: Discrete-Time Signals and Systems the output of the system is given by either of the following two equivale ...

8.3 Discrete-Time Systems 489 or the convolution sum of the input and the impulse response. This is a very important result, ind ...

490 C H A P T E R 8: Discrete-Time Signals and Systems FIGURE 8.9 (a) Cascade and (b) parallel connections of LTI systems with i ...

8.3 Discrete-Time Systems 491 Now ifx[n] is the input to the filter according to the convolution sum, its output is y[n]= ∑n k= ...

492 C H A P T E R 8: Discrete-Time Signals and Systems FIGURE 8.10 Convolution sums for a moving-averaging filter with input x 1 ...

8.3 Discrete-Time Systems 493 from which a general expression for the impulse response is obtained ash[n]=0.5nu[n]. The response ...

494 C H A P T E R 8: Discrete-Time Signals and Systems FIGURE 8.11 First-order autoregressive filter impulse responseh[n] and re ...

8.3 Discrete-Time Systems 495 ThisM-order filter averagesMpast input values{y[n−k],k=0,...,M− 1 }and assigns this average to the ...

496 C H A P T E R 8: Discrete-Time Signals and Systems 0 20 40 60 80 100 (a) (b) 120 140 160 180 0 20 40 60 80 100 120 140 160 1 ...

8.3 Discrete-Time Systems 497 0 20 40 60 80 100 (a) 120140160180 200 − 2 − 1 0 1 2 n y^1 [n ] (b) (c) 0 20 40 60 80 100 120 140 ...

498 C H A P T E R 8: Discrete-Time Signals and Systems situations, real-time processing is not required as the data can be store ...

8.3 Discrete-Time Systems 499 to signals we can then see that the output of a causal LTI discrete-time system can be written in ...

500 C H A P T E R 8: Discrete-Time Signals and Systems For the given case wherey[n]=δ[n] andx[n]=u[n], we get, according to the ...

8.3 Discrete-Time Systems 501 An LTI discrete-time system is said to be BIBO stable if its impulse responseh[n]is absolutely sum ...

502 C H A P T E R 8: Discrete-Time Signals and Systems 8.4 What Have We Accomplished? Where Do We Go from Here?................. ...

Problems 503 or equivalently that ( 1 −α) ∑∞ n= 0 αn= 1 provided|α|< 1. (c) Verify your results by using symbolic MATLAB to f ...