Given 5.0||)5.0(4)( 2 >+= zzzzX Find the Z-Transform of the following signals.(a) y[n] = x[ n – 2] (b) e[n] = x[–n] (c) d[n] = 2n x[n] (d) s[n] = x[n] * x[n]A N S :4(a) 2)5.0z(zz4)z(Y += |Z| > 0.5 (b) 2)2(16)(+=zzE (c)1||:)1(8)( 2 >+=

To find the Z-Transform of the given signals, we will follow these steps:

(a) For y[n] = x[n – 2]:

• Apply the time-shifting property of the Z-Transform, which states that if x[n] has a Z-Transform X(z), then x[n – k] has a Z-Transform z^(-k)X(z).
• Substitute n – 2 for n in the Z-Transform of x[n].
• The Z-Transform of y[n] is 2z^(-2)X(z).

(b) For e[n] = x[–n]:

• Apply the time-reversal property of the Z-Transform, which states that if x[n] has a Z-Transform X(z), then x[–n] has a Z-Transform X(1/z).
• Substitute –n for n in the Z-Transform of x[n].
• The Z-Transform of e[n] is X(1/z).

(c) For d[n] = 2n x[n]:

• Apply the scaling property of the Z-Transform, which states that if x[n] has a Z-Transform X(z), then nx[n] has a Z-Transform –z(dX(z)/dz).
• Take the derivative of X(z) with respect to z.
• Multiply the derivative by –z and substitute 2n for n.
• The Z-Transform of d[n] is –2z(dX(z)/dz).

(d) For s[n] = x[n] * x[n]:

• Apply the convolution property of the Z-Transform, which states that if x[n] has a Z-Transform X(z) and y[n] has a Z-Transform Y(z), then the convolution of x[n] and y[n] has a Z-Transform X(z)Y(z).
• Multiply the Z-Transform of x[n] by itself.
• The Z-Transform of s[n] is X(z)^2.

Therefore, the Z-Transforms of the given signals are: (a) Y(z) = 2z^(-2)X(z) (b) E(z) = X(1/z) (c) D(z) = –2z(dX(z)/dz) (d) S(z) = X(z)^2