Given H(z)=(1z1)(1z1) Where =ej4
a) Find H(ej)
b) This transfer function has zeros on the unit circle at z=e±j4, what does that tell you about the amplitude response function H(ej) when =4
The correct answer and explanation is :
Problem Breakdown and Analysis
Given the transfer function:
[
H(z) = (1 + z^{-1})(1 + z^{-1})
]
This can be rewritten in a more manageable form for evaluation, recognizing that it’s the product of two identical terms:
[
H(z) = (1 + z^{-1})^2
]
We’re tasked with finding the evaluation of ( H(z) ) on the unit circle, where ( z = e^{j\omega} ), for the specific value ( \omega = 4 ).
a) Find ( H(e^{j\omega}) )
Let’s evaluate ( H(z) ) when ( z = e^{j\omega} ):
[
H(e^{j\omega}) = \left( 1 + e^{-j\omega} \right)^2
]
Substitute ( \omega = 4 ):
[
H(e^{j4}) = \left( 1 + e^{-j4} \right)^2
]
Now we simplify this expression:
- ( e^{-j4} ) is the complex exponential with ( e^{-j4} = \cos(4) – j\sin(4) ).
- So, ( H(e^{j4}) = (1 + \cos(4) – j\sin(4))^2 ).
Next, expand this squared term:
[
H(e^{j4}) = \left( 1 + \cos(4) – j\sin(4) \right)^2
]
Apply the binomial expansion:
[
H(e^{j4}) = (1 + \cos(4))^2 – 2j(1 + \cos(4))\sin(4) – \sin^2(4)
]
This expression gives the value of ( H(e^{j4}) ) in terms of real and imaginary components.
b) Analysis of Zeros on the Unit Circle and the Amplitude Response
We are told that the transfer function has zeros at ( z = e^{\pm j4} ). This is crucial for understanding the amplitude response of ( H(e^{j\omega}) ).
- Zeros on the Unit Circle: When the zeros of a transfer function lie on the unit circle in the z-plane (i.e., at ( z = e^{j\omega} ) for some ( \omega )), it means that the transfer function has zeros at frequencies where the magnitude of the frequency response goes to zero. In other words, the response at these frequencies will be completely attenuated.
- Amplitude Response: The amplitude response is defined as ( |H(e^{j\omega})| ). Since the zeros are located at ( z = e^{\pm j4} ), this implies that at ( \omega = 4 ), the transfer function will have a zero. Therefore, the magnitude of the frequency response ( |H(e^{j4})| ) will be zero at this frequency.
- What does this tell us?: When ( \omega = 4 ), the amplitude response ( |H(e^{j4})| ) is zero due to the presence of a zero at ( z = e^{j4} ). This means that at this frequency, the output will be completely attenuated, or “nullified.” Hence, for ( \omega = 4 ), the system effectively “rejects” any input at that frequency. This result is consistent with the idea that a system’s frequency response can be zeroed out at specific frequencies where zeros of the transfer function occur.
Conclusion
In summary:
- a) ( H(e^{j4}) ) is evaluated as ( (1 + e^{-j4})^2 ), which is a complex expression involving trigonometric functions of ( 4 ).
- b) The zeros of the transfer function at ( z = e^{\pm j4} ) imply that the amplitude response at ( \omega = 4 ) will be zero, meaning the system rejects signals at this frequency. This highlights how the locations of zeros influence the frequency response, especially at points where zeros lie on the unit circle.