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How is the frequency response of an ideal differentiator related to the frequency?
Explanation: An ideal differentiator has a frequency response that is linearly proportional to the frequency.
Which of the following is the frequency response of an ideal differentiator, Hd(ω)?
Explanation: An ideal differentiator is defined as one that has the frequency response
H_{d}(ω)= jω ; π ≤ ω ≤ π.
What is the unit sample response corresponding to Hd(ω)?
Explanation: We know that, for an ideal differentiator, the frequency response is given as
H_{d}(ω)= jω ; π ≤ ω ≤ π
Thus, we get the unit sample response corresponding to the ideal differentiator is given as
h(n)=cosπn/n.
The ideal differentiator ahs which of the following unit sample response?
Explanation: We know that the unit sample response of an ideal differentiator is given as
h(n)=cosπn/n
So, we can state that the unit sample response of an ideal differentiator is antisymmetric because cosπn is also an antisymmetric function.
If h_{d}(n) is the unit sample response of an ideal differentiator, then what is the value of h_{d}(0)?
Explanation: Since we know that the unit sample response of an ideal differentiator is antisymmetric,
=>h_{d}(0)=0.
In this section, we confine our attention to FIR designs in which h(n)=h(M1n).
Explanation: In view of the fact that the ideal differentiator has an antisymmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=h(M1n).
Which of the following is the condition that an differentiator should satisfy?
Explanation: For an FIR filter, when M is odd, the real valued frequency response of the FIR filter Hr(ω) has the characteristic that Hr(0)=0. A zero response at zero frequency is just the condition that the differentiator should satisfy.
Full band differentiators can be achieved with an FIR filters having odd number of coefficients.
Explanation: Full band differentiators cannot be achieved with an FIR filters having odd number of coefficients, since Hr(π)=0 for M odd.
If f_{p} is the bandwidth of the differentiator, then the desired frequency characteristic should be linear in the range:
Explanation: In most cases of practical interest, the desired frequency response characteristic need only be linear over the limited frequency range 0 ≤ ω ≤ 2πf_{p} , where f_{p} is the bandwidth of the differentiator.
What is the desired response of the differentiator in the frequency range 2πf_{p} ≤ ω ≤ π?
Explanation: In the frequency range 2πf_{p} ≤ ω ≤ π, the desired response may be either left unconstrained or constrained to be zero.
What is the weighting function used in the design of FIR differentiators based on the chebyshev approximation criterion?
Explanation: In the design of FIR differentiators based on the chebyshev approximation criterion, the weighting function W(ω) is specified in the program as
W(ω)=1/ω
in order that the relative ripple in the pass band be a constant.
The absolute error between the desired response ω and the approximation H_{r}(ω) decreases as ω varies from 0 to 2πf_{p}.
Explanation: We know that the weighting function is
W(ω)=1/ω
in order that the relative ripple in the pass band be a constant. Thus, the absolute error between the desired response ω and the approximation H_{r}(ω) increases as ω varies from 0 to 2πf_{p}.
Which of the following is the important parameter in a differentiator?
Explanation: The important parameters in a differentiator are its length, its bandwidth and the peak relative error of the approximation. The inter relationship among these three parameters can be easily displayed parametrically.
In this section, we confine our attention to FIR designs in which h(n)=h(M1n).
Explanation: In view of the fact that the ideal differentiator has an antisymmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=h(M1n).
What is the maximum value of f_{p} with which good designs are obtained for M odd?
Explanation: Designs based on M odd are particularly poor if the bandwidth exceeds 0.45. The problem is basically the zero in the frequency response at ω=π(f=1/2). When f_{p} <0.45, good designs are obtained for M odd.
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