Low-Delay Band-Pass Maximally Flat FIR Digital Differentiators

Abstract

The present paper describes a closed-form transfer function of low-delay band-pass maximally flat FIR digital differentiators. Band-pass maximally flat FIR digital differentiators provide extremely high-accuracy differentiation around the center frequency which is adjusted arbitrarily. At the same time, they can reduce noise in the frequency except around the center frequency. However, the conventional method for designing band-pass maximally flat FIR digital differentiators requires linear phase characteristics. In contrast, the proposed method can realize low-delay characteristics as well as linear phase characteristics and, therefore, is a general expression of band-pass maximally flat FIR digital differentiators. The proposed transfer function is achieved as the sum of two maximally flat complex FIR digital differentiators, the coefficients of which are complex conjugates of each other. The transfer functions of these complex differentiators are derived as closed-form solutions, so that the proposed transfer function is also described as a closed-form solution. Through design examples, the effectiveness of the proposed method is confirmed.

Proof that (7) satisfies (2)

1.1 Proof for (2a)–(2e)

We consider the following equation:

$$\begin{aligned}&{j\omega e^{-j\omega \tau }-P_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)}\nonumber \\&\quad =(E^{-j\omega }+1)^{2\kappa }(e^{-j\omega }-1)^{2\mu }(e^{-j\omega }-e^{j\omega _0})^{L+1}\Bigl [\Bigl \{j\omega e^{-j\omega \tau }(e^{-j\omega }+1)^{-2\kappa }\nonumber \\&\quad \cdot (e^{-j\omega }-1)^{-2\mu }(e^{-j\omega }-e^{j\omega _0})^{-(L+1)}\Bigr \}-Q_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)\Bigr ]\nonumber \\&\quad =(e^{-j\omega }+1)^{2\kappa }(e^{-j\omega }-1) ^{2\mu }(e^{-j\omega }-e^{j\omega _0})^{L+1}\nonumber \\&\quad \left\{ Q_{\infty , \tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)- Q_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)\right\} , \end{aligned}$$

(10)

where

$$\begin{aligned}&{Q_{\infty ,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)}\nonumber \\&\quad =\sum _{n=0}^{\infty }\frac{1}{n!}\frac{d^n}{d\zeta }\nonumber \\&\quad \left\{ -\ln (\zeta ) \zeta ^{\tau }(\zeta +1)^{-2\kappa }\left. (\zeta -1)^{-2\mu }(\zeta -\zeta _0^{-1})^{-(L+1)}\right\} \right| _{\zeta =\zeta _0}\nonumber \\&\quad \cdot (e^{-j\omega }-e^{-j\omega _0})^n\nonumber \\&\quad =\sum _{n=0}^{\infty }q_{n,\tau }^{\kappa ,\mu }(\omega _0)(e^{-j\omega }-e^{-j\omega _0})^{n},\nonumber \\&\quad {\left\{ \begin{array}{ll} \zeta =e^{-j\omega }\\ \zeta _0=e^{-j\omega _0}\\ -\ln (\zeta )=j\omega . \end{array}\right. } \end{aligned}$$

(11)

Using (10) and (11), $P_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)$ is rewritten as

$$\begin{aligned} P_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)= & {} j\omega e^{-j(\omega \tau - \theta _0)}-(e^{-j\omega }-e^{-j\omega _0})^{L+1}\varPsi _1 (e^{j\omega }, \omega _0),\quad \end{aligned}$$

(12)

where

$$\begin{aligned} \varPsi _1 (e^{j\omega },\omega _0)= & {} (e^{-j\omega }+1)^{2\kappa }(e^{-j\omega }-1)^{2\mu }(e^{-j\omega }-e^{j\omega _0})^{L+1}\nonumber \\&\cdot \sum _{n=0}^{\infty }q_{n+L+1,\tau }^{\kappa ,\mu }(\omega _0)(e^{-j\omega }-e^{-j\omega _0})^{n}, \end{aligned}$$

(13)

and $\varPsi _1 (e^{j\omega },\omega _0)$ is bounded at $\omega =\omega _0$.

From (7) and (12), $H(e^{j\omega })$ can be deformed as

$$\begin{aligned} H(e^{j\omega })= & {} j\omega e^{-j(\omega \tau -\theta _0)}-(e^{-j\omega }-e^{-j\omega _0})^{L+1}\varPsi _2 (e^{j\omega }, \omega _0)\nonumber \\= & {} j\omega e^{-j(\omega \tau -\theta _0)}\Bigl \{ 1-(\omega -\,\omega _0)^{L+1}\varPsi _3 (e^{j\omega }, \omega _0)\Bigr \}, \end{aligned}$$

(14)

where

$$\begin{aligned} \varPsi _2 (e^{j\omega }, \omega _0)= & {} e^{j\theta _0}\varPsi _1 (e^{j\omega },\omega _0)\nonumber \\&-\,e^{-j\theta _0}(e^{-j\omega }+1)^{2\kappa }(e^{-j\omega }-1)^{2\mu }Q_{L,\tau }^{\kappa ,\mu }(e^{j\omega },-\,\omega _0) \end{aligned}$$

(15)

$$\begin{aligned} \varPsi _3 (e^{j\omega }, \omega _0)= & {} \frac{(-je^{-j\omega _0})^{L+1}}{j\omega e^{-j\omega \tau }}\left\{ \frac{e^{-j(\omega -\,\omega _0)}-1}{-j(\omega -\,\omega _0)}\right\} ^{L+1}\varPsi _2 (e^{j\omega }, \omega _0), \end{aligned}$$

(16)

and both $\varPsi _2 (e^{j\omega },\omega _0)$ and $\varPsi _3 (e^{j\omega },\omega _0)$ are bounded at $\omega =\omega _0$.

Next, we assume the transfer function of the low-delay low-pass MFDDs [35] given by

$$\begin{aligned} H_{low}(e^{j\omega })= & {} j\omega e^{-j(\omega \tau )}\Bigl \{ 1-(\omega )^{2\mu +1}{\tilde{\varPsi }} (e^{j\omega })\Bigr \}, \end{aligned}$$

(17)

where ${\tilde{\varPsi }}(e^{j\omega })$ is the function bounded at $\omega =0$. In [35], $H_{\mathrm{low}}(e^{j\omega })$ was proven to satisfy both the magnitude and the group delay constraints at $\omega =0$. Thus, the proof that (14) satisfies both the magnitude and group delay constraints at $\omega =\omega _0$ can be provided by applying the proof in the previous study [35]. Moreover, (7) is also expressed as

$$\begin{aligned}&H(e^{j\omega })=j\omega e^{-j(\omega \tau +\theta _0)}\Bigl \{ 1-(\omega +\omega _0)^{L+1}\varPsi _3 (e^{j\omega }, -\,\omega _0)\Bigr \}, \end{aligned}$$

(18)

so that the proof for the constraints at $\omega =-\,\omega _0$ can be provided in the same manner as the proof at $\omega =\omega _0$. Therefore, it is clear that (7) satisfies (2a)–(2e).$\square$

1.2 Proof for (2f) and (2g)

Equation (7) can be transformed into

$$\begin{aligned}&H(e^{j\omega })=(\cos \omega +1)^{\kappa }(\cos \omega -1)^{\mu }\varPhi (e^{j\omega }), \end{aligned}$$

(19)

where $\varPhi (e^{j\omega })$ is bounded at $\omega =\{0, \pi \}$ and is given by

$$\begin{aligned} \varPhi (e^{j\omega })= & {} \Big (\frac{e^{-j\omega }}{2}\Big )^{\kappa + \mu }\left\{ (e^{-j\omega }-e^{j\omega _0})^{L+1}Q_{L,\tau }^{\kappa ,\mu }(e^{j\omega },\omega _0)\right. \nonumber \\&\left. +(e^{-j\omega }-e^{-j\omega _0})^{L+1}Q_{L,\tau }^{\kappa ,\mu }(e^{j\omega },-\,\omega _0)\right\} , \end{aligned}$$

(20)

Thus, (19) satisfies (2f) and (2g) according to the proof for the magnitude constraints of linear phase band-pass MFDDs at $\omega =\{0, \pi \}$ [32].$\square$