Test: FIR Differentiator Design - Electronics and Communication Engineering (ECE) MCQ

Explanation: An ideal differentiator has a frequency response that is linearly proportional to the frequency.

Explanation: An ideal differentiator is defined as one that has the frequency response
H_d(ω)= jω ; -π ≤ ω ≤ π.

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.

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 anti-symmetric because cos⁡πn is also an anti-symmetric function.

Explanation: Since we know that the unit sample response of an ideal differentiator is anti-symmetric,
=>h_d(0)=0.

Explanation: In view of the fact that the ideal differentiator has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

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.

Explanation: Full band differentiators cannot be achieved with an FIR filters having odd number of coefficients, since Hr(π)=0 for M odd.

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.

Explanation: In the frequency range 2πf_p ≤ ω ≤ π, the desired response may be either left unconstrained or constrained to be zero.

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.

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.

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.

Explanation: In view of the fact that the ideal differentiator has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

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.