16.7. All-Pass Filter Design

Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

16.7. All-Pass Filter Design

In comparison to the previously discussed filters, an all-pass filter has a constant gain across the entire frequency range, and a phase response that changes linearly with frequency.

Because of these properties, all-pass filters are used in phase compensation and signal delay circuits.

Similar to the low-pass filters, all-pass circuits of higher order consist of cascaded first-order and second-order all-pass stages. To develop the all-pass transfer function from a low-pass response, replace A₀ with the conjugate complex denominator.

The general transfer function of an all-pass is

$A (s) = \frac{\prod_{i} (1 - a_{i} s + b_{i} s^{2})}{\prod_{i} (1 + a_{i} s + b_{i} s^{2})}$ $A (s) = \frac{\prod_{i} (1 - a_{i} s + b_{i} s^{2})}{\prod_{i} (1 + a_{i} s + b_{i} s^{2})}$

(16.23)

with a_i and b_i being the coefficients of a partial filter. The all-pass coefficients are listed in Table 16.12 of Section 16.9.

Expressing Eq. (16.23) in magnitude and phase yields

$A (s) = \frac{\prod_{i} \sqrt{{(1 - b_{i} Ω^{2})}^{2} + a_{i}^{2} Ω^{2} \cdot e^{- ja}}}{\prod_{i} \sqrt{{(1 - b_{i} Ω^{2})}^{2} + a_{i}^{2} Ω^{2} \cdot e^{+ ja}}}$ $A (s) = \frac{\prod_{i} \sqrt{{(1 - b_{i} Ω^{2})}^{2} + a_{i}^{2} Ω^{2} \cdot e^{- ja}}}{\prod_{i} \sqrt{{(1 - b_{i} Ω^{2})}^{2} + a_{i}^{2} Ω^{2} \cdot e^{+ ja}}}$

(16.24)

This gives a constant gain of 1, and a phase shift,φ, of:

$φ = - 2 α = - 2 \sum_{i} \arctan \frac{a_{i} Ω}{1 - b_{i} Ω^{2}}$ $φ = - 2 α = - 2 \sum_{i} \arctan \frac{a_{i} Ω}{1 - b_{i} Ω^{2}}$

(16.25)

To transmit a signal with minimum phase distortion, the all-pass filter must have a constant group delay across the specified frequency band. The group delay is the time by which the all-pass filter delays each frequency within that band.

The frequency at which the group delay drops to

1 / \sqrt{2}

$1 / \sqrt{2}$

times its initial value is the corner frequency, f_C.

The group delay is defined through:

$t_{gr} = - \frac{d φ}{d ω}$ $t_{gr} = - \frac{d φ}{d ω}$

(16.26)

To present the group delay in normalized form, refer t_gr to the period of the corner frequency, T_C, of the all-pass circuit:

$T_{gr} = \frac{t_{gr}}{T_{c}} = t_{gr} \cdot f_{c} = t_{gr} \cdot \frac{ω_{c}}{2 π}$ $T_{gr} = \frac{t_{gr}}{T_{c}} = t_{gr} \cdot f_{c} = t_{gr} \cdot \frac{ω_{c}}{2 π}$

(16.27)

Substituting t_gr through Eq. (16.26) gives

$T_{gr} = - \frac{1}{2 π} \cdot \frac{d φ}{d Ω}$ $T_{gr} = - \frac{1}{2 π} \cdot \frac{d φ}{d Ω}$

(16.28)

Inserting the φ term in Eq. (16.25) into Eq. (16.28) and completing the derivation, results in:

$T_{gr} = \frac{1}{π} \sum_{i} \frac{a_{i} (1 + b_{i} Ω^{2})}{1 + (a_{1}^{2} - 2 b_{1}) \cdot Ω^{2} + b_{1}^{2} Ω^{4}}$ $T_{gr} = \frac{1}{π} \sum_{i} \frac{a_{i} (1 + b_{i} Ω^{2})}{1 + (a_{1}^{2} - 2 b_{1}) \cdot Ω^{2} + b_{1}^{2} Ω^{4}}$

(16.29)

Setting Ω = 0 in Eq. (16.29) gives the group delay for the low frequencies, 0 < Ω< 1, which is

$T_{gr 0} = \frac{1}{π} \sum_{i} a_{i}$ $T_{gr 0} = \frac{1}{π} \sum_{i} a_{i}$

(16.30)

The values for T_gr0 are listed in Table 16.12, Section 16.9, from the first to the tenth order.

In addition, Fig. 16.42 shows the group delay response versus the frequency for the first ten orders of all-pass filters.

Figure 16.42 Frequency response of the group delay for the first 10 filter orders.

16.7.1. First-Order All-Pass Filter

Fig. 16.43 shows a first-order all-pass filter with a gain of +1 at low frequencies and a gain of −1 at high frequencies. Therefore, the magnitude of the gain is 1, while the phase changes from 0 degrees to −180 degrees.

The transfer function of the circuit above is

$A (s) = \frac{1 - RC ω_{c} \cdot s}{1 + RC ω_{c} \cdot s}$ $A (s) = \frac{1 - RC ω_{c} \cdot s}{1 + RC ω_{c} \cdot s}$

The coefficient comparison with Eq. (16.23) (b1 = 1) results in:

$a_{i} = RC \cdot 2 π f_{c}$ $a_{i} = RC \cdot 2 π f_{c}$

(16.31)

To design a first-order all-pass, specify f_C and C and then solve for R:

$R = \frac{a_{i}}{2 π f_{c} \cdot C}$ $R = \frac{a_{i}}{2 π f_{c} \cdot C}$

(16.32)

Inserting Eq. (16.31) into Eq. (16.30) and substituting ω_C with Eq. (16.27) provides the maximum group delay of a first-order all-pass filter:

$t_{gr 0} = 2 RC$ $t_{gr 0} = 2 RC$

(16.33)

16.7.2. Second-Order All-Pass Filter

Fig. 16.44 shows that one possible design for a second-order all-pass filter is to subtract the output voltage of a second-order band-pass filter from its input voltage.

The transfer function of the circuit in Fig. 16.44 is

$A (s) = \frac{1 + (2 R_{1} - α R_{2}) C ω_{c} \times s + R_{1} R_{2} C^{2} ω_{c}^{2} \times s^{2}}{1 + 2 R_{1} C ω_{c} \times s + R_{1} R_{2} C^{2} ω_{c}^{2} \times s^{2}}$ $A (s) = \frac{1 + (2 R_{1} - α R_{2}) C ω_{c} \times s + R_{1} R_{2} C^{2} ω_{c}^{2} \times s^{2}}{1 + 2 R_{1} C ω_{c} \times s + R_{1} R_{2} C^{2} ω_{c}^{2} \times s^{2}}$

Figure 16.44 Second-order all-pass filter.

The coefficient comparison with Eq. (16.23) yields

$a_{1} = 4 π f_{c} R_{1} C$ $a_{1} = 4 π f_{c} R_{1} C$

(16.34)

$b_{1} = a_{1} π f_{c} R_{2} C$ $b_{1} = a_{1} π f_{c} R_{2} C$

(16.35)

$α = \frac{a_{1}^{2}}{b_{1}} = \frac{R}{R_{3}}$ $α = \frac{a_{1}^{2}}{b_{1}} = \frac{R}{R_{3}}$

(16.36)

To design the circuit, specify f_C, C, and R, and then solve for the resistor values:

$R_{1} = \frac{a_{1}}{4 π f_{c} C}$ $R_{1} = \frac{a_{1}}{4 π f_{c} C}$

(16.37)

$R_{2} = \frac{b_{1}}{a_{1} π f_{c} C}$ $R_{2} = \frac{b_{1}}{a_{1} π f_{c} C}$

(16.38)

$R_{3} = \frac{R}{α}$ $R_{3} = \frac{R}{α}$

(16.39)

Inserting Eq. (16.34) into Eq. (16.30) and substituting ω_C with Eq. (16.27) gives the maximum group delay of a second-order all-pass filter:

$t_{gr 0} = 4 R_{1} C$ $t_{gr 0} = 4 R_{1} C$

(16.40)

16.7.3. Higher-Order All-Pass Filter

Higher-order all-pass filters consist of cascaded first-order and second-order filter stages.

Example 16.7. 2-ms Delay All-Pass Filter.

A signal with the frequency spectrum, 0 < f < 1 kHz, needs to be delayed by 2 ms. To keep the phase distortions at a minimum, the corner frequency of the all-pass filter must be f_C ≥ 1 kHz.

Eq. (16.27) determines the normalized group delay for frequencies below 1 kHz:

$T_{gr 0} = \frac{t_{gr 0}}{T_{c}} = 2 ms \cdot 1 kHz = 2.0$ $T_{gr 0} = \frac{t_{gr 0}}{T_{c}} = 2 ms \cdot 1 kHz = 2.0$

Fig. 16.42 confirms that a seventh-order all-pass is needed to accomplish the desired delay. The exact value, however, is T_gr0 = 2.1737. To set the group delay to precisely 2 ms, solve Eq. (16.27) for f_C and obtain the corner frequency:

$f_{C} = \frac{T_{gr 0}}{t_{gr 0}} = 1.087 kHz$ $f_{C} = \frac{T_{gr 0}}{t_{gr 0}} = 1.087 kHz$

To complete the design, look up the filter coefficients for a seventh-order all-pass filter, specify C, and calculate the resistor values for each partial filter.

Cascading the first-order all-pass with the three second-order stages results in the desired seventh-order all-pass filter (See Fig. 16.45).

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for 16.7. All-Pass Filter Design

Create new playlist

Sign In

Sign Up

16.7. All-Pass Filter Design

16.7.1. First-Order All-Pass Filter

16.7.2. Second-Order All-Pass Filter

16.7.3. Higher-Order All-Pass Filter

Table of Contents for
16.7. All-Pass Filter Design