Digital Filters Design Consideration MCQ [Free PDF] – Objective Question Answer for Digital Filters Design Consideration Quiz

If the HI(ω) is uniquely determined from HR(ω) through the integral relationship. This integral is called a discrete Hilbert transform.

Causality has very important implications in the design of frequency-selective filters. One among them is the magnitude |H(ω)| cannot be constant in any finite range of frequencies and the transition from pass-band to stop-band cannot be infinitely sharp. This is a consequence of the Gibbs phenomenon, which results from the truncation of h(n) to achieve causality.

Passband edge ripple is the frequency at which the passband starts transiting to the stopband.

If ωP and ωS represent the passband edge ripple and stopband edge ripple, then the transition width -ωP+ ωS gives the bandwidth of the filter.

An FIR filter of length M with input x(n) and output y(n) is described by the difference equation

y(n)=\(\sum_{k=0}^{M-1} b_k x(n-k)\)
where {bk} is the set of filter coefficients.

We can express the output sequence as the convolution of the unit sample response h(n) of the system with the input signal. The lower and upper limits on the convolution sum reflect the causality and finite duration characteristics of the filter.

An FIR filter has an linear phase if its unit sample response satisfies the condition
h(n)= ±h(M-1-n) n=0,1,2…M-1.

We know that H(z)=\(\sum_{k=0}^{M-1} h(k)z^{-k}\) and h(n)=±h(M-1-n) n=0,1,2…M-1

When we incorporate the symmetric and anti-symmetric conditions of the second equation into the first equation and by substituting z-1 for z and multiplying both sides of the resulting equation by z-(M-1) we get z-(M-1).H(z-1)=±H(z)

We know that z-(M-1).H(z-1)=±H(z). This result implies that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1).

We know that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1). Consequently, the roots of H(z) must occur in reciprocal pairs.

We know that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1). This implies that if the unit sample response h(n) of the filter is real, complex-valued roots must occur in complex conjugate pairs.

When h(n)=-h(M-1-n), the unit sample response is anti-symmetric. For M odd, the center point of the anti-symmetric is n=M-1/2. Consequently, h(M-1/2)=0.

We know that, for a symmetric h(n), the number of filter coefficients that specify the frequency response is (M+1)/2 when M is odd and M/2 when M is even.