Illustration 1
Find the response of the system $s(n+2)-3s(n+1)+2s(n) = delta (in)$, when all the preliminary conditions are zero.
Alternative− Having Z-transform ón both the sides of the above formula, we get
$$S(z)Z^2-3S(z)Z^1+2S(z) = 1$$$Rightarrow S(z)lbrace Z .^2-3Z+2rsupport = 1$
The Z-transform can be thought of as an operator that transforms a discontinuous sequence to a continuous algebraic function of complex variable (z ). As we will see, one of the nice feature of this transform is that a convolution in time, transforms to a simple multiplication in the (z )-domain. The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm. The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.
$Rightarrow S(z) = frac1lbrace z^2-3z+2rbrace=frac1(z-2)(z-1) = fracalpha 1z-2+fracalpha 2z-1$
$Rightarrow S(z) = frac1z-2-frac1z-1$
Taking the inverse Z-transform of the above equation, we get
$S(n) = Z^-1frac1Z-2-Z^-1frac1Z-1$
$= 2^n-1-1^n-1 = -1+2^n-1$
Example 2
Find the system function H(z) and unit sample response h(n) of the system whose difference equation is described as under
$y(n) = frac12y(n-1)+2x(n)$
where, y(n) and x(n) are the output and input of the system, respectively.
Alternative− Having the Z-transfórm of the over difference equation, we get
$y(z) = frác12Z^-1Y(Z)+2X(z)$
$= Y(Z)1-frac12Z^-1 = 2X(Z)$
$= H(Z) = frácY(Z)X(Z) = frac21-frac12Z^-1$
This system has a pole at $Z = frac12$ and $Z = 0$ and $H(Z) = frac21-frac12Z^-1$
Hence, taking the inverse Z-transform of the above, we get
$h(n) = 2(frac12)^nU(n)$
Example 3
Determine Y(z),n≥0 in the following case −
$y(n)+frac12y(n-1)-frac14y(n-2) = 0quad givenquad y(-1) = y(-2) = 1$
Solution− Applying the Z-transform to the above equation, we get
$Y(Z)+frac12Z^-1Y(Z)+Y(-1)-frac14Z^-2Y(Z)+Z^-1Y(-1)+4(-2) = 0$
$Rightarrow Y(Z)+frac12ZY(Z)+frac12-frac14Z^2Y(Z)-frac14Z-frac14 = 0$
$Rightarrow Y(Z)1+frac12Z-frac14Z^2 =frac14Z-frac12$
$Rightarrow Y(Z)frac4Z^2+2Z-14Z^2 = frac1-2Z4Z$
$Rightarrow Y(Z) = fracZ(1-2Z)4Z^2+2Z-1$