Signals And Systems

Step 1: Start with the definition of the transfer function. The transfer function

H(z)

relates the Z-transform of the output

Y(z)

to the Z-transform of the input

X(z)

H(z)%20%3D%20%5Cfrac%7BY(z)%7D%7BX(z)%7D

Given

H(z)%20%3D%20%5Cfrac%7B1%7D%7B1%20-%20%5Cfrac%7B1%7D%7B4%7Dz%5E%7B-1%7D%7D

, we can write:

%5Cfrac%7BY(z)%7D%7BX(z)%7D%20%3D%20%5Cfrac%7B1%7D%7B1%20-%20%5Cfrac%7B1%7D%7B4%7Dz%5E%7B-1%7D%7D

Step 2: Rearrange the equation to isolate input and output terms. Cross-multiply to get rid of the fraction:

Y(z)%20%5Cleft(%201%20-%20%5Cfrac%7B1%7D%7B4%7Dz%5E%7B-1%7D%20%5Cright)%20%3D%20X(z)

Distribute

Y(z)

on the left side:

Y(z)%20-%20%5Cfrac%7B1%7D%7B4%7Dz%5E%7B-1%7DY(z)%20%3D%20X(z)

Step 3: Apply the inverse Z-transform. We use the time-shifting property of the Z-transform, which states that

%5Cmathcal%7BZ%7D%5E%7B-1%7D%5C%7Bz%5E%7B-k%7DF(z)%5C%7D%20%3D%20f%5Bn-k%5D

. Taking the inverse Z-transform of each term in the equation:

%5Cmathcal%7BZ%7D%5E%7B-1%7D%5C%7BY(z)%5C%7D%20%5Crightarrow%20y%5Bn%5D

%5Cmathcal%7BZ%7D%5E%7B-1%7D%5C%7B%5Cfrac%7B1%7D%7B4%7Dz%5E%7B-1%7DY(z)%5C%7D%20%5Crightarrow%20%5Cfrac%7B1%7D%7B4%7Dy%5Bn-1%5D

%5Cmathcal%7BZ%7D%5E%7B-1%7D%5C%7BX(z)%5C%7D%20%5Crightarrow%20x%5Bn%5D

Combining these gives the difference equation:

y%5Bn%5D%20-%20%5Cfrac%7B1%7D%7B4%7Dy%5Bn-1%5D%20%3D%20x%5Bn%5D

Step 1: Test for Linearity.
A system is linear if it satisfies superposition. Let $y_1%5Bn%5D$ be the output for input $x_1%5Bn%5D$ and $y_2%5Bn%5D$ for $x_2%5Bn%5D$ . For an input $x%5Bn%5D%20%3D%20a%20x_1%5Bn%5D%20%2B%20b%20x_2%5Bn%5D$ , the output is $y%5Bn%5D$ .
For $n%20%5Cge%201$ , $y%5Bn%5D%20%3D%20x%5Bn%5D%20%3D%20a%20x_1%5Bn%5D%20%2B%20b%20x_2%5Bn%5D%20%3D%20a%20y_1%5Bn%5D%20%2B%20b%20y_2%5Bn%5D$ .
For $n%20%5Cle%20-1$ , $y%5Bn%5D%20%3D%20x%5Bn%2B1%5D%20%3D%20a%20x_1%5Bn%2B1%5D%20%2B%20b%20x_2%5Bn%2B1%5D%20%3D%20a%20y_1%5Bn%5D%20%2B%20b%20y_2%5Bn%5D$ .
The system is composed of linear operations on the input, so it is linear.

Step 2: Test for Time-Invariance.
A system is time-invariant if a shift in the input causes an identical shift in the output. Let $y'%5Bn%5D$ be the response to $x%5Bn-d%5D$ . We must check if $y'%5Bn%5D%20%3D%20y%5Bn-d%5D$ .
Let's find the output for $x%5Bn-1%5D$ (i.e., d=1): $y'%5Bn%5D%20%3D%20%5Cbegin%7Bcases%7D%20x%5Bn-1%5D%2C%20%26%20n%20%5Cge%201%20%5C%5C%200%2C%20%26%20n%3D0%20%5C%5C%20x%5Bn%5D%2C%20%26%20n%20%5Cle%20-1%20%5Cend%7Bcases%7D$ .
Now let's find the shifted original output $y%5Bn-1%5D$ : Replace $n$ with $n-1$ in the definition of $y%5Bn%5D$ .
$y%5Bn-1%5D%20%3D%20%5Cbegin%7Bcases%7D%20x%5Bn-1%5D%2C%20%26%20n-1%20%5Cge%201%20%5Cimplies%20n%20%5Cge%202%20%5C%5C%200%2C%20%26%20n-1%3D0%20%5Cimplies%20n%3D1%20%5C%5C%20x%5Bn%5D%2C%20%26%20n-1%20%5Cle%20-1%20%5Cimplies%20n%20%5Cle%200%20%5Cend%7Bcases%7D$ .
Comparing $y'%5Bn%5D$ and $y%5Bn-1%5D$ , they are not identical (e.g., at $n%3D1$ , $y'%5B1%5D%3Dx%5B0%5D$ while $y%5B1-1%5D%3Dy%5B0%5D%3D0$ ). The system is not time-invariant.

Step 3: Test for Causality.
A system is causal if the output $y%5Bn%5D$ depends only on the current and past inputs $x%5Bk%5D$ where $k%20%5Cle%20n$ .
For $n%20%5Cle%20-1$ , the output is $y%5Bn%5D%20%3D%20x%5Bn%2B1%5D$ . Since $n%2B1%20%3E%20n$ , the output depends on a future value of the input. Therefore, the system is not causal.

Step 4: Test for Stability (BIBO).
A system is Bounded-Input, Bounded-Output (BIBO) stable if every bounded input produces a bounded output. Assume the input is bounded, i.e., $%7Cx%5Bn%5D%7C%20%5Cle%20M_x%20%3C%20%5Cinfty$ for all $n$ .
For $n%20%5Cge%201$ , $%7Cy%5Bn%5D%7C%20%3D%20%7Cx%5Bn%5D%7C%20%5Cle%20M_x$ .
For $n%3D0$ , $%7Cy%5B0%5D%7C%20%3D%200$ .
For $n%20%5Cle%20-1$ , $%7Cy%5Bn%5D%7C%20%3D%20%7Cx%5Bn%2B1%5D%7C%20%5Cle%20M_x$ .
In all cases, the output is bounded by $M_x$ . The system is stable.

Conclusion: The system is Linear (A) and Stable (D).

About Signals And Systems - CUET-PG

Signals And Systems is a vital chapter for CUET-PG aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

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Signals And Systems

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Chapter Questions
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