FIR Least Squares Inverse Filter MCQ [Free PDF] – Objective Question Answer for FIR Least Squares Inverse Filter Quiz

FIR least-square filters are also called as Wiener filters.

The inverse to a linear time-invariant system with impulse response h(n) is defined as the system whose impulse response is h_I(n), satisfying the following condition h(n)*h_I(n)=δ(n).

The inverse to a linear time-invariant system with impulse response h(n) and system function H(z) is defined as the system whose impulse response is h_I(n) and system function H_I(z), satisfies the following condition

H(z).H_I(z)=1.

In most practical applications, it is desirable to restrict the inverse filter to be an FIR filter.

In many practical applications, it is desirable to restrict the inverse filter to FIR. One of the simple methods to get this requirement is to truncate h_I(n).

In the process of truncating, we incur a total squared approximation error where M+1 is the length of the truncated filter.

We can use the least-squares error criterion to optimize the M+1 coefficients of the FIR filter.

Since from the block diagram, the coefficients of the FIR filter coefficients are optimized by the least-squares error criterion, it belongs to the least-squares FIR inverse filter or wiener filter.

When ε is minimized with respect to the filter coefficients, we obtain the set of linear equations which are dependent on the autocorrelation sequence of the signal h(n).

When ε is minimized with respect to the filter coefficients, we obtain the set of linear equations
\(\sum_{k=0}^M b_k r_{hh} (k-l)=r_{dh} (l)\), l=0,1,…M

and we know that r_dh(l) depends on the desired output d(n).

The optimum, in the least square sense, FIR filter that satisfies the linear equations in \(\sum_{k=0}^M b_k r_{hh} (k-l)=r_{dh} (l)\) =r_{dh} (l)), l=0,1,…M is called the wiener filter.

If the optimum least-squares FIR filter is to be an approximate inverse filter, the desired response is
d(n)=δ(n).

We observe that the matrix is not only symmetric but it also has the special property that all the elements along any diagonal are equal. Such a matrix is called a Toeplitz matrix and lends itself to efficient inversion by means of an algorithm.

The Levinson-Durbin algorithm is the algorithm that is used for the efficient inversion of the Toeplitz matrix which requires a number of computations proportional to M² instead of the usual M³.

We describe a filter parameter optimization technique carried out in the frequency domain that is representative of frequency domain design methods.

The design is most easily carried out with the system function for the IIR filter expressed in the cascade form as
H(z)=G.A(z).

Instead of dealing with the phase response ϴ(ω), it is more convenient to deal with the envelope delay as a function of frequency.

Instead of dealing with the phase response ϴ(ω), it is more convenient to deal with the envelope delay as a function of frequency, which is

T_g(ω)= -dϴ(ω)/dω.

The error in magnitude at the frequency ω_k is G.A(ω_k) – A_d(ω_k) for 0 ≤ |ω| ≤ π, where A_d(ω_k) is the desired magnitude response at ω_k.

Similarly, as in the previous question, the error in delay at ω_k is defined as T_g(ω_k)-T_d(ω_k), where T_d(ω_k) is the desired delay response.

We know that the error in the delay is defined as T_g(ω_k) – T_d(ω_k). However, the choice of T_d(ω_k) for error in the delay is complicated by the difficulty in assigning a nominal delay of the filter.

We are led to define the error in delay as T_g(ω_k) – T_g(ω₀) – T_d(ω_k), where T_g(ω₀) is the filter delay at some nominal center frequency in the passband of the filter.

As a performance index for determining the filter parameters, one can choose any arbitrary function of the errors in magnitude and delay.

In the error function, ‘p’ denotes the 4K dimension vector of the filter coefficients.

The emphasis on the errors affecting the design may be placed entirely on the delay by taking the value of λ as 1.

The emphasis on the errors affecting the design may be equally weighted between magnitude and delay by taking the value of λ as 1/2.

The squared error function E(p,G) is a non-linear function of 4K+1 parameters.

Due to the non-linear nature of E(p,G), its minimization over the remaining 4K parameters is performed by an iterative numerical optimization method.

The major difficulty with any iterative procedure that searches for the parameter values that minimize a non-linear function is that the process may converge to a local minimum instead of a global minimum.