FIR filter design using DFT matrix inversion

Question

In my previous question I use a system identification case as an example. Here I came across another question. I want to design a FIR filter of length $L$ to equalize an LTI system whose impulse response is $h(n)$ of length $N$, and my target response is $d(n)$ which is maybe a delayed $\delta(n-D)$. The problem can be formulated as finding the unknown $g(n)$ that satisfies $$ h(n)*g(n)=d(n) $$ Frequency domain deconvolution can be used. Calculate $N+L-1$ point FFT $$ H(k) \cdot G(k) = D(k) $$ Impose regularization we have $$ G(k) = \frac{D(k)H(k)^*}{H(k)H(k)^*+\epsilon(k)} $$

Now I have to transform $G(k)$ back to time domain by $N+L-1$ point IFFT. But I can't guarantee that the IFFT result will end up with $L-1$ point zeros. I can simply truncate the result as @Hilmar suggests, but I don't think this is mathematically correct.

I came up with a solution with the DFT matrix. Let $$ \mathbf{g}= \big[g(0), g(1), \ldots, g(N-1)\big]^T $$ be the time-domain vector of $g(n)$, and $$ \mathbf{G} = \big[G(0), G(1), \ldots, G(N+L-2)\big]^T $$ be the frequency-domain vector of $G(k)$, the DFT matrix $T$ represents the linear transform $$ \mathbf{Tg=G} $$ However, due to $g(n) $ and $G(k)$ have different lengths, this DFT matrix is not a square matrix, it is a truncated DFT matrix.

Finally, the FIR coefficients can be derived by the least-squares method $$ \mathbf{g}=\big(\mathbf{T}^H\mathbf{T}+\lambda \mathbf{I}\big)^{-1}\mathbf{T}^H \mathbf{G} $$

Since the final solution is a least-square solution, I can't guarantee that $N+L-1$ point FFT of $g(n)$ equals to $G(k)$.

Is it a good way to design FIR filters? Does it have any disadvantage? Any better way?

Answer 1

Given that we have information already in the time domain, do the least squares solution in that domain, which is the approach used for least squares equalizers: This works quite well to correct for a linear channel distortion when the sounding signal uses has rich spectral occupancy (since the accuracy of the solution for any given point in frequency will be proportional to the SNR for that point). I suspect (but haven't confirmed) that equalizing the channel in the frequency domain using the DFT would suffer the same drawbacks as using the "Frequency Sampling Method" for FIR filter design, which is inferior due to time domain aliasing as demonstrated at this link.

The graphic below shows the solution we seek: we know the sounding sequence $t[n]$ and the received output of the channel $r[n]$. We desire a corrected $r[n]$ to match as close as possible the original sounding sequence, using the equalizer filter with coefficients $h[n]$. The corrected sequence at the output of the equalizer is shown as $t'[n]$, which we desire $t'[n]=t[n]$ after allowable delay.

The output of the equalizer FIR is the convolution of $r[n]$ with the equalizer coefficients $h[n]$. Convolution can be written as a matrix equation by using $r[n]$ in a Toeplitz matrix as demonstrated in the graphic below.

Thus the resulting matrix equation is given as:

$$t' = A \cdot h$$

We wish to solve for $h$, but $A$ is not a square matrix so can't be inverted. We can multiply $A$ by its conjugate transpose to make it invertible, multiplying both sides to not change the equation, resulting in the classical form for the least squares solution to an overdetermined equation:

$$A^Tt' = A^TA \cdot h$$

$$h = (A^TA)^{-1}A^T t'$$

I further detail this including MATLAB code and how intentional delay is importantly included at this link.

The same formulation can instead be used to estimate the channel rather than invert it as demonstrated in this link.