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For apply least-squares linear-phase FIR filter design,with frequency domain specification is not symmetrical.

The pass-band error function,

$$E(\mathbf{h})_p=\int_{\omega_{p_1}}^{\omega_{p_2}}| \mathbf{c}^T(\omega)\cdot \mathbf{h}-D(\omega)|^2d\omega \tag{1}$$

The stop-band error function,

$$E(\mathbf{h})_s=\int_{\omega_{s_1}}^{\omega_{s_2}}|\mathbf{c}^T(\omega)\cdot \mathbf{h}|^2d\omega\tag{2}$$

h: unknown complex coefficients.

Total error function,

$$E(\mathbf{h})_t=w_1E(\mathbf{h})_p+w_2E(\mathbf{h})_s\tag{3}$$

$w_1,w_2$: weighting constants .

  • The equation $(1)$ : which represents a filter approximation by determine the error between actual and desired response in pass-band , why the method is different in stop-band (equation $(2)$) ?

-In matlab ,does it the freqz function achieve the transpose operation ?

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In the stopband(s), Eqs $(1)$ and $(2)$ are equivalent, because in the stopband the desired response equals zero: $D(\omega)=0$. The reason why you might want to split the error in passband and stopband error is to apply different weights, as shown in your Eq. $(3)$.

The function freqz computes the frequency response of a discrete-time filter on a grid of frequencies, given the filter coefficients. I'm not sure what you mean by "achieve the transpose operation".

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  • $\begingroup$ Matt L:Why there is no Hermitian operation ,why there is just transpose ?why using it? $\endgroup$ Commented Sep 14, 2018 at 10:33
  • $\begingroup$ @K.n90: It's up to you, depending on how you define the vector $\mathbf{c}(\omega)$. Add your definition of $\mathbf{c}(\omega)$, and it will become clear if you need to use $^H$ or $^T$ (or none of the two). $\endgroup$ Commented Sep 14, 2018 at 10:38
  • $\begingroup$ :The definition in this answer [] (dsp.stackexchange.com/a/50937/32272), is what I need but without using Hermitian operation. In other hand, does the choice of the coefficients type (h) have an effect on using the operations ( Hermitian ,transpose)? $\endgroup$ Commented Sep 14, 2018 at 11:16
  • $\begingroup$ @K.n90: If you want to use the definition used in the answer you mentioned, you have to use $^H$ (as shown there), otherwise use your own definition. I don't understand the problem. $\endgroup$ Commented Sep 14, 2018 at 11:19

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