Digital DC blocker filters

Klaus Tittelbach-Helmrich

doi:10.1515/freq-2020-0177

Article Open Access

Digital DC blocker filters

Klaus Tittelbach-Helmrich

Published/Copyright: February 18, 2021

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From the journal Frequenz Volume 75 Issue 9-10

Abstract

This paper mathematically investigates a special kind of digital infinite-impulse response (IIR) filters, suitable for filtering out very low frequencies near zero from digital signals. We investigate the transfer functions of such filters from 1st to 3rd order and provide formulas to calculate the filter coefficients from the desired cutoff frequency.

Keywords: DC blocker; digital filter; IIR filter

1 Introduction

Digital filtering plays a very important role in many data processing systems. Sometimes it is necessary to remove a time-invariant additive offset from a signal [1], [2]. Such signal components are often named “DC offset”, the respective filter “DC blocker” (DC = “Directed Current” in contrast to “Alternating Current” = AC). For example, Analog-to-Digital converters (ADCs) often possess such a small constant offset that needs compensation or removal [2], [3]. The required filter is a high-pass filter that lets uniformly pass all signal frequencies except extremely low ones, say few percent of the sampling rate or even less.

This paper investigates a special kind of digital DC blocker filters. We provide formulas to calculate the filter coefficients when the desired corner frequency is known. The filter structure can be implemented in software or digital hardware.

2 Digital filter basics

A digital filter is a hardware or software unit that processes an (infinite) series of digital input samples {x _k}, producing a series of output samples {y _k}. Each output sample y _k is a linear combination of a number M of preceding input samples x _k−m (m = 0–M) and (optionally) a number N of previous output samples y _k−n (n = 1–N). If N = 0, we call it a “Finite Impulse Response” (FIR) filter, since a singular non-zero pulse at the input will lead to a finite number of output pulses (at most M of them). If N > 0, the number of emerging output pulses may theoretically be infinite. This is the “Infinite-Impulse Response” (IIR) filter. It may possess some complications such as insufficient stability, but normally the desired spectral shaping of the output signal, i.e., the filtering, can be achieved with less computational effort than in FIR filters.

Mathematically, the filter is described by its equation

(1) y k = ∑ m = 0 M b m x k − m + ∑ n = 1 N a n y k − n

where the x _k−m are the input samples, y _k−n the output samples, and the b _m and a _n are constant real coefficients. Their values determine the characteristics of the filter, i.e., the spectral shaping of the signal when it passes the filter.

The samples are almost always taken at constant intervals in time Δt, or, in other words, at a constant sampling frequency f s = 1 / Δ t . The input and output samples can then be written as x _k = x(k ⋅ Δt) and y _k = y(k ⋅ Δt), respectively. Equation (1) then transforms to

(2) y ( k ⋅ Δ t ) = ∑ m = 0 M b m x ( ( k − m ) ⋅ Δ t ) + ∑ n = 1 N a y n ( ( k − n ) ⋅ Δ t )

For investigating the spectral properties, we perform a Fourier transform (in the following we only show important final results, for more details consult a textbook like [4], [5], [6]).

(3) Y ( Ω ) = X ( Ω ) ∑ m = 0 M b m e − i Ω m + Y ( Ω ) ∑ n = 1 N a n e − i Ω n

Here, X(Ω) and Y(Ω) are the Fourier-transformed input and output signals. Ω is the normalized angular frequency. We normalize all frequencies to the sampling frequency f _S.

(4) Ω = 2 π ⋅ f ⋅ Δ t = 2 π ⋅ f / f S

According to the Nyquist theorem, the highest frequency component in the signal, which makes sense, is the Nyquist frequency f _N = ½ f _S. Thus, the sensible range for Ω is from 0 to π = 180° = Nyquist frequency. The sampling frequency f _S would correspond to Ω = 2π.

The ratio of the output signal to the input signal in frequency domain is denoted “transfer function” H(Ω).

(5) H ( Ω ) = Y ( Ω ) X ( Ω ) = ∑ m = 0 M b m e − i Ω ⋅ m 1 − ∑ n = 1 N a n e − i Ω ⋅ n

The transfer function is a complex function of the (normalized) frequency, i.e., it has a real and an imaginary part, or an absolute value and a phase. The squared absolute value | H ( Ω ) | 2 is the most important feature of the filter since it describes its spectral properties, i.e., which frequency components are attenuated or amplified. Also, the phase φ(Ω) is sometimes considered, but we will not discuss this property in this paper. The absolute value computes as:

(6) | H ( Ω ) | 2 = ∑ m = 0 M b m 2 + 2 ⋅ ∑ m = 1 M ( cos ( Ω ⋅ m ) ⋅ ( ∑ k = m M b k b k − m ) ) 1 + ∑ n = 1 N a n 2 − 2 ⋅ ∑ n = 1 N ( cos ( Ω ⋅ n ) ⋅ ( a n − ∑ k = n + 1 N a k a k − n ) )

Finally, we investigate the stability of the filter. This means, that the filter output samples {y _k} are and remain finite (limited) for any finite input sequence {x _k}. In particular, they must remain finite after a single input pulse. Mathematically, instability occurs at poles of the transfer function H(Ω). For investigating this property, we transform (5) into the z domain by substituting e ^iΩ = z:

(7) H ( Ω ) = ∑ m = 0 M b m e − i Ω ⋅ m 1 − ∑ n = 1 N a n e − i Ω ⋅ n = ∑ m = 0 M b m z − m 1 − ∑ n = 1 N a n z − n

The poles occur when the denominator is zero, i.e.,

(8) 1 − ∑ n = 1 N a n z − n = 0 or z N − ∑ n = 1 N a n z N − n = 0

The filter is stable when all complex poles z _k of (7), equivalent to the polynomial roots of (8), are inside the unit circle of the z domain or, in other words, all | z k | < 1 [6].

The stability of a filter depends only on the recursive (feedback) coefficients a _n. FIR filters, where all a _n = 0, are always stable.

3 DC blocker filters

The DC blocker filter shall block very low frequencies near zero and let all higher signal components pass unchanged. This is the high-pass behavior. The transfer function shall be H(Ω) = 0 for Ω = 0 and H(Ω) = 1 for Ω = π (Nyquist frequency).

3.1 First order

The textbook DC blocker filter ([4] chapter 13.23, [2]) is a 1st order IIR filter having the transfer function

(1.1) H ( Ω ) = 1 − e − i Ω 1 − α ⋅ e − i Ω ⋅

This corresponds to the filter equation

(1.2) y k = ( x k − x k − 1 ) + α ⋅ y k − 1

Parameter α determines the corner frequency. For the DC blocker it is slightly less than one, say in the range 0.95–0.99.

We will now derive these equations from our general formulas in Section 2. Naming and normalization will slightly differ from the conventions in (1.1) and (1.2). For example: from (1.1) we deduce a pass-band gain of H(Ω = π) = 2/(1 + α); in our normalization we get H(Ω = π) = 1 for any high-pass filter.

Inserting M = N = 1 into (5) leads to

(1.3) H ( Ω ) = b 0 + b 1 ⋅ e − i Ω 1 − a 1 ⋅ e − i Ω

For a high-pass filter we set the boundary conditions H(Ω = 0) = 0 (frequency zero shall be blocked) and H(Ω = π) = 1 (the Nyquist frequency component shall pass unchanged). This allows elimination of two of the unknown coefficients b ₀, b ₁ and a ₁.

From the first condition we get with e ^–i0 = 1:

(1.4) H ( Ω = 0 ) = b 0 + b 1 1 − a 1 = 0

or, in other words, b ₁ = −b ₀ (provided that a ₁ ≠ 1).

From the second condition we get with e ^–iπ = −1:

(1.5) H ( Ω = π ) = b 0 − b 1 1 + a 1 = 1

or, in other words, b ₀ − b ₁ = 1 + a ₁. We redefine the one remaining free parameter as b in the following way:

(1.6) b 0 = b b 1 = − b a 1 = b 0 − b 1 − 1 = 2 b − 1

We insert this into (6):

(1.7) | H ( Ω ) | 2 = b 0 2 + b 1 2 + 2 ⋅ b 0 b 1 ⋅ cos ( Ω ) 1 + a 1 2 − 2 ⋅ a 1 ⋅ cos ( Ω ) = b 2 + b 2 − 2 b 2 ⋅ cos ( Ω ) 1 + ( 2 b − 1 ) 2 − 2 ⋅ ( 2 b − 1 ) ⋅ cos ( Ω ) = 2 b 2 ⋅ ( 1 − cos ( Ω ) ) 4 b 2 − 4 b ⋅ ( 1 + cos ( Ω ) ) + 2 ⋅ ( 1 + cos ( Ω ) )

We transform this equation using the trigonometric relations (1 − cos(Ω)) = 2⋅sin²(¹/₂ ⋅ Ω), (1 + cos(Ω)) = 2⋅cos²(¹/₂ ⋅ Ω), and sin²(¹/₂ ⋅ Ω) + cos²(¹/₂ ⋅ Ω) = 1.

(1.8) | H ( Ω ) | 2 = 4 b 2 ⋅ sin 2 ( ½ Ω ) 4 b 2 + 4 ( 1 − 2 b ) ⋅ cos 2 ( ½ Ω ) = b 2 ⋅ sin 2 ( ½ Ω ) b 2 ⋅ ( sin 2 ( ½ Ω ) + cos 2 ( ½ Ω ) ) + ( 1 − 2 b ) ⋅ cos 2 ( ½ Ω ) = b 2 ⋅ sin 2 ( ½ Ω ) b 2 ⋅ sin 2 ( ½ Ω ) + ( 1 − 2 b + b 2 ) ⋅ cos 2 ( ½ Ω ) | H ( Ω ) | 2 = 1 1 + ( 1 − b ) 2 ⋅ cos 2 ( ½ Ω ) b 2 ⋅ sin 2 ( ½ Ω )

We re-formulate the free parameter again as b = 1 − ¹/₂ ω or ω = 2(1 − b) (we will see the reason later):

(1.9) | H ( Ω ) | 2 = 1 1 + ( ω 2 − ω ⋅ cot ( Ω 2 ) ) 2

For Ω in the range 0 ≤ Ω ≤ π, |H(Ω)|² is a monotonic function. Figure 1 shows some graphs. The larger ω, the higher the corner frequency of the filter, i.e., the transition point from blocking to passing behavior. In case ω = 0 Equation (1.9) has a singularity at Ω = 0.

Figure 1:

Transfer function of 1st order DC blocker filters (Equation [1.9]).

A common criterion for the corner frequency is the point, where the filter’s attenuation reaches 3 dB, i.e., |H(Ω)|² = ½. We calculate this point from (1.9):

| H ( Ω 3 dB ) | 2 = 1 2 = 1 1 + ( ω 2 − ω ⋅ cot ( Ω 3 dB 2 ) ) 2

( ω 2 − ω ⋅ cot ( Ω 3 dB 2 ) ) 2 = 1

(1.10) tan ( Ω 3 dB 2 ) = ± ω 2 − ω

For a DC blocker, the corner frequency shall be low: Ω_3dB << 1. In this case, we apply the series expansion for tan(x) = x − 1/3 x ³ + … Also ω will be small, so that ω << 2.

(1.11) Ω 3 dB ≈ 2 ⋅ ω 2 − ω ≈ ω

Our parameter ω equals the (normalized, angular) 3 dB corner frequency of the filter. If the latter is given, we can calculate the filter coefficients b ₀, b ₁ and a ₁ (see (1.6) and text above (1.9)):

a 1 = 1 − Ω 3 dB

b 0 = + ( 1 − ½ Ω 3 dB )

(1.12) b 1 = − ( 1 − ½ Ω 3 dB )

These equations are valid only for small corner frequencies. Equation (1.10), however, applies to any frequency.

Finally, we verify the stability of the filter, which is not always certain for IIR filters. For doing this, we have to find the poles of (1.3). Inserting (1.12), the equation reads as:

H ( Ω ) = b 0 + b 1 e − i Ω 1 − a 1 e − i Ω ⋅ = ( 1 − ½ ω ) ⋅ ( 1 − e − i Ω ) 1 − ( 1 − ω ) ⋅ e − i Ω ⋅

We substitute e ^iΩ = z for transforming into the z domain:

H ( Ω ) = ( 1 − ½ ω ) ⋅ ( 1 − z − 1 ) 1 − ( 1 − ω ) ⋅ z − 1 = ( 1 − ½ ω ) ⋅ ( z − 1 ) z − ( 1 − ω )

The pole occurs when the denominator is zero, i.e.,

z − ( 1 − ω ) = 0 or z = 1 − ω

In the z domain, the filter is stable when |z| < 1 [6]. This is the case when 0 < ω < 2, which is fulfilled for a DC blocker filter. However, the nearer ω to 0 (i.e., the smaller the corner frequency), the more the filter approaches the stability limit.

3.2 Second order

Figure 1 depicts that the first order filters are not very steep. For many applications, a smaller transition range between stop and pass behavior is highly desired. This can be achieved by increasing the order of the filter, i.e., the number of coefficients.

For a 2nd order filter we have to set M = N = 2 in (5)

(2.1) H ( Ω ) = b 0 + b 1 ⋅ e − i Ω + b 2 ⋅ e − i 2 Ω 1 − a 1 ⋅ e − i Ω − a 2 ⋅ e − i 2 Ω

The boundary conditions for high-pass filter do not change: H(Ω = 0) = 0 (frequency zero shall be blocked) and H(Ω = π) = 1 (Nyquist frequency shall pass unchanged). This again allows elimination of two of the unknown coefficients, but we have five of them now: b ₀, b ₁, b ₂ and a ₁, a ₂.

From the first condition we get with e ⁻ⁱ⁰ = 1:

(2.2) H ( Ω = 0 ) = b 0 + b 1 + b 2 1 − a 1 − a 2 = 0

or, in other words, b ₁ = −(b ₀ + b ₂) (provided that a ₁ + a ₂ ≠ 1).

From the second condition we get:

(2.3) H ( Ω = π ) = b 0 − b 1 + b 2 1 + a 1 − a 2 = 1

or, in other words, b ₀ − b ₁ + b ₂ = 1 + a ₁ − a ₂. Combining this with the result from (2.2) leads to 2(b ₀ + b ₂) = −2b ₁ = 1 + a ₁ − a ₂ or a ₁ = 2(b ₀ + b ₂) + a ₂ − 1. Inserting into (6) yields

(2.4) | H ( Ω ) | 2 = b 0 2 + ( b 0 + b 2 ) 2 + b 2 2 − 2 ( b 0 + b 2 ) 2 ⋅ cos ( Ω ) + 2 b 0 b 2 ⋅ cos ( 2 ⋅ Ω ) 1 + ( 2 ( b 0 + b 2 ) + a 2 − 1 ) 2 + a 2 2 − 2 ( 2 ( b 0 + b 2 ) + a 2 − 1 ) ⋅ ( 1 − a 2 ) ⋅ cos ( Ω ) − 2 a 2 ⋅ cos ( 2 ⋅ Ω )

We transform this equation in a way similar to (1.6). Only main intermediate results are given.

(2.5) | H ( Ω ) | 2 = 2 ( b 0 + b 2 ) 2 ⋅ ( 1 − cos ( Ω ) ) − 2 b 0 b 2 ⋅ ( 1 − cos ( 2 Ω ) ) 1 + a 2 2 + ( a 2 − 1 ) 2 + 4 ( b 0 + b 2 ) 2 + 4 ( b 0 + b 2 ) ⋅ ( a 2 − 1 ) + 2 ( 2 ( b 0 + b 2 ) + a 2 − 1 ) ⋅ ( a 2 − 1 ) cos ( Ω ) − 2 a 2 cos ( 2 Ω ) = 4 ( b 0 + b 2 ) 2 ⋅ sin 2 ( Ω 2 ) − 4 b 0 b 2 ⋅ ( 1 − cos 2 ( Ω ) ) 2 ( a 2 − 1 ) 2 + 4 ( b 0 + b 2 ) 2 + 8 ( b 0 + b 2 ) ⋅ ( a 2 − 1 ) ⋅ cos 2 ( Ω 2 ) + 2 ( a 2 − 1 ) 2 ⋅ cos ( Ω ) + 2 a 2 ⋅ ( 1 − cos ( 2 ⋅ Ω ) ) = 4 ( b 0 + b 2 ) 2 ⋅ sin 2 ( Ω 2 ) − 4 b 0 b 2 ⋅ ( 1 − cos ( Ω ) ) ⋅ ( 1 + cos ( Ω ) ) 4 ( b 0 + b 2 ) 2 + 8 ( b 0 + b 2 ) ⋅ ( a 2 − 1 ) ⋅ cos 2 ( Ω 2 ) + 2 ( a 2 − 1 ) 2 ⋅ ( 1 + cos ( Ω ) ) + 4 a 2 ( 1 − cos ( Ω ) ) ⋅ ( 1 + cos ( Ω ) ) = 4 ( b 0 + b 2 ) 2 ⋅ sin 2 ( Ω 2 ) − 16 ⋅ b 0 b 2 ⋅ sin 2 ( Ω 2 ) ⋅ ( 1 − sin 2 ( Ω 2 ) ) 4 ( ( b 0 + b 2 ) 2 + 4 a 2 ) ⋅ sin 2 ( Ω 2 ) + 4 ( ( b 0 + b 2 ) 2 + ( 2 ( b 0 + b 2 ) + ( a 2 − 1 ) ) ⋅ ( a 2 − 1 ) ) ⋅ cos 2 ( Ω 2 ) − 16 a 2 sin 4 ( Ω 2 ) = 4 ( b 0 − b 2 ) 2 ⋅ sin 2 ( Ω 2 ) + 16 ⋅ b 0 b 2 ⋅ sin 4 ( Ω 2 ) 4 ( ( b 0 + b 2 ) 2 + 4 a 2 ) ⋅ sin 2 ( Ω 2 ) + 4 ( ( b 0 + b 2 ) + ( a 2 − 1 ) ) 2 ⋅ cos 2 ( Ω 2 ) − 16 a 2 sin 4 ( Ω 2 ) = ( b 0 − b 2 ) 2 ⋅ sin 2 ( Ω 2 ) + 4 ⋅ b 0 b 2 ⋅ sin 4 ( Ω 2 ) ( b 0 − b 2 ) 2 ⋅ sin 2 ( Ω 2 ) + 4 ( b 0 b 2 + a 2 ) ⋅ sin 2 ( Ω 2 ) − 4 a 2 ⋅ sin 4 ( Ω 2 ) + ( ( b 0 + b 2 ) + ( a 2 − 1 ) ) 2 ⋅ cos 2 ( Ω 2 )

This formula already looks pretty similar to the 1st order transfer function (1.8), provided that b ₀ = b ₂ and a ₂ = −b ₀ ⋅ b ₂. It will turn out that this setting is indeed a good solution. Thus, we may set our remaining free parameters as b ₀ = b ₂ = b and a ₂ = −b ₀ ⋅ b ₂ = −b ².

(2.6) | H ( Ω ) | 2 = 4 b 2 ⋅ sin 4 ( Ω 2 ) 4 b 2 ⋅ sin 4 ( Ω 2 ) + ( 2 b − b 2 − 1 ) 2 ⋅ cos 2 ( Ω 2 ) = 1 1 + ( ( 1 − b ) 2 2 b ⋅ sin ( Ω / 2 ) ⋅ cot ( Ω 2 ) ) 2

More generally, we may set b ₀ = b + β, b ₂ = b − β and a ₂ = α − b ₀ ⋅ b ₂ = α − (b ² − β ²). Inserting into (2.5) leads to:

(2.7) | H ( Ω ) | 2 = 4 β 2 ⋅ sin 2 ( Ω 2 ) + 4 ⋅ ( b 2 − β 2 ) ⋅ sin 4 ( Ω 2 ) 4 β 2 ⋅ sin 2 ( Ω 2 ) + 4 α ⋅ sin 2 ( Ω 2 ) − 4 α ⋅ sin 4 ( Ω 2 ) + 4 ( b 2 − β 2 ) ⋅ sin 4 ( Ω 2 ) + ( α + β 2 − ( 1 − 2 b + b 2 ) ) 2 ⋅ cos 2 ( Ω 2 ) = 4 β 2 ⋅ sin 2 ( Ω 2 ) ⋅ cos 2 ( Ω 2 ) + 4 b 2 ⋅ sin 4 ( Ω 2 ) 4 β 2 ⋅ sin 2 ( Ω 2 ) ⋅ cos 2 ( Ω 2 ) + 4 b 2 ⋅ sin 4 ( Ω 2 ) + 4 α ⋅ sin 2 ( Ω 2 ) ⋅ cos 2 ( Ω 2 ) + ( α + β 2 − ( 1 − b ) 2 ) 2 ⋅ cos 2 ( Ω 2 ) = 1 1 + 4 α ⋅ sin 2 ( Ω 2 ) + ( ( 1 − b ) 2 − α − β 2 ) 2 4 β 2 ⋅ cos 2 ( Ω 2 ) + 4 b 2 ⋅ sin 2 ( Ω 2 ) ⋅ cot 2 ( Ω 2 )

Comparing (2.6) and (2.7) one finds that a small non-zero β is almost equivalent to a change in b, i.e., to a shift of the corner frequency. Moreover, only β ² is present in (2.7), i.e., the sign of β is irrelevant. In contrast, a negative α can lead to |H(Ω)|² > 1 when the term 4α ⋅ sin²(¹/₂⋅Ω) + ((1−b)² − α − β ²)² becomes negative.

Figure 2 shows some numerical simulations. It turns out that α = 0 is in fact a good choice. Positive α leads to flatter curve shape of the transfer function, whereas a negative α may result in a peak of |H(Ω)|². Both is not desired. A slightly improved curve shape may be obtained for small negative α ≈ −(1−b)²/8 = −ω ²/16.

Coming back to our simple case (2.6), we re-define the parameter b that determines the corner frequency as ω = √2 (1 − b). The 2nd order Formula (2.8) is now very similar to the 1st order Formula (1.9), but it has an additional factor ω/2 sin(¹/₂ ⋅ Ω) before the cot() term. This term is responsible for the steeper transfer curve of the 2nd order filter, compared to the 1st order one.

(2.8) | H ( Ω ) | 2 = 1 1 + ( ω 2 − 2 ⋅ ω ⋅ ω 2 ⋅ sin ( Ω / 2 ) ⋅ cot ( Ω 2 ) ) 2

For Ω in the range 0 ≤ Ω ≤ π, |H(Ω)|² is a monotonic function. Figure 3 shows some graphs. The larger ω, the higher the corner frequency of the filter, i.e., the transition from blocking to passing behavior. In the case ω = 0 Equation (2.8) has a singularity at Ω = 0.

Figure 2:

Influence of non-zero α and β in Equation (2.7).

Figure 3:

Transfer function of 2nd order DC blocker filters (Equation [2.8]).

We calculate the 3 dB corner frequency from (2.8):

| H ( Ω 3 dB ) | 2 = 1 2 = 1 1 + ( ω 2 4 − 8 ⋅ ω ⋅ cot ( ½ Ω 3 dB ) sin ( ½ Ω 3 dB ) ) 2

( ω 2 4 − 8 ⋅ ω ⋅ cot ( ½ Ω 3 dB ) sin ( ½ Ω 3 dB ) ) 2 = 1

(2.9) tan ( ½ Ω 3 dB ) ⋅ sin ( ½ Ω 3 dB ) = ± ( ω 2 4 − 8 ⋅ ω )

For a DC blocker, the corner frequency shall be low: Ω_3dB << 1. In this case, we apply the series expansion for tan(x) = x − 1/3 x ³ + … and sin(x) = x − 1/6 x ³ + … Also ω will be small, so that 8 ω << 4.

(2.10) Ω 3 dB ⋅ Ω 3 dB ≈ 4 ⋅ ω 2 4 − 8 ⋅ ω ≈ ω 2

Our parameter ω again equals the (normalized, angular) 3 dB corner frequency of the filter. If the latter is given, we can calculate the filter coefficients:

a 1 = 3 − ( 1 + ½ ⋅ Ω 3 dB ) 2 = 2 ⋅ ( 1 − ½ ⋅ Ω 3 dB ) − ½ ⋅ Ω 3 dB 2

a 2 = − ( 1 − ½ ⋅ Ω 3 dB ) 2

b 0 = 1 − ½ ⋅ Ω 3 dB

b 1 = − 2 ⋅ ( 1 − ½ ⋅ Ω 3 dB )

(2.11) b 2 = 1 − ½ ⋅ Ω 3 dB

These equations are valid only for small corner frequencies. Equation (2.8), however, applies to any frequency.

Finally, we verify the stability of the filter. We have to find the poles of (2.1). Inserting (2.10), the z-transformed equation reads as:

H ( Ω ) = ( 1 − ½ ⋅ ω ) ⋅ ( z 2 − 2 z + 1 ) z 2 − ( 3 − ( 1 + ½ ⋅ ω ) 2 ) z + ( 1 − ½ ⋅ ω ) 2

The poles occur when the denominator is zero, i.e.,

z 2 − 2 ⋅ ( 1 − ½ ⋅ ω − 1 4 ⋅ ω 2 ) z + ( 1 − ½ ⋅ ω ) 2 = 0

This is a quadratic polynomial. Its roots are given by Vieta’s formula:

z = ( 1 − ½ ⋅ ω − 1 4 ⋅ ω 2 ) ± ( 1 − ½ ⋅ ω − ¼ ⋅ ω 2 ) 2 − ( 1 − ½ ⋅ ω ) 2 = ( 1 − ½ ⋅ ω − ¼ ⋅ ω 2 ) ± 1 16 ⋅ ω 4 + ⅛ ⋅ ω 3 − ½ ⋅ ω 2

For small ω << 1 the term under the square root is negative, i.e., the two poles are conjugate complex. Their absolute value is given by the sum of the squares of real and imaginary parts:

| z | 2 = ( 1 − ½ ⋅ ω − ¼ ω 2 ) 2 + ( ½ ⋅ ω 2 − ⅛ ⋅ ω 3 − 1 16 ω 4 ) = 1 + ½ ω 2 + 1 16 ω 4 − 2 ⋅ ω − ½ ω 2 + ⅛ ⋅ ω 3 + ½ ⋅ ω 2 − ⅛ ⋅ ω 3 − 1 16 ω 4 = 1 − 2 ⋅ ω + ½ ⋅ ω 2

| z | 2 = ( 1 − ½ ⋅ ω ) 2

The filter is stable when |z| < 1. This is the case when 0 < ω < 2 2 , which is fulfilled for a DC blocker filter. However, the nearer ω to 0 (i.e., the smaller the corner frequency), the more the filter approaches the stability limit.

3.3 Third order

In order to further increase the steepness of the transfer function, we may further increase the order of the filter. We will provide here the results for the 3rd order filter, without giving all intermediate steps and results. The way of calculations is similar to the 2nd order filter.

For a 3rd order filter we have to set M = N = 3 in (5)

(3.1) H ( Ω ) = b 0 + b 1 ⋅ e − i Ω + b 2 ⋅ e − i 2 Ω + a 3 ⋅ e − i 3 Ω 1 − a 1 ⋅ e − i Ω − a 2 ⋅ e − i 2 Ω − a 3 ⋅ e − i 3 Ω

The boundary conditions for high-pass filter do not change: H(Ω = 0) = 0 (frequency zero shall be blocked) and H(Ω = π) = 1 (Nyquist frequency shall pass unchanged). This again allows elimination of two of the unknown coefficients, but we now have seven of them: b ₀, b ₁, b ₂, b ₃ and a ₁, a ₂, a ₃.

From these boundary conditions we get:

(3.2) H ( Ω = 0 ) = b 0 + b 1 + b 2 + b 3 1 − a 1 − a 2 − a 3 = 0

(3.3) H ( Ω = π ) = b 0 − b 1 + b 2 − b 3 1 + a 1 − a 2 + a 3 = 1

The calculations proceed in the same way as for 2nd order. Since we have two more free coefficients, we need more criteria to pre-select some of them. On one-hand side, we have chosen the b _k coefficients to behave like binomial coefficients (i.e., to have a relation b ₀ = b, b ₁ = −3b, b ₂ = 3b, b ₃ = −b) in concordance with the lower order filter’s b _k coefficients. On the other hand, the target |H(Ω)|² transfer function should have only one cot²(Ω/2) term. The calculations are quite lengthy, so we report only the final results. The transfer function reads as:

(3.4) | H ( Ω ) | 2 = 1 1 + ( ω 3 4 ⋅ ( 1 − ω ) ⋅ ( 2 − ω ) ⋅ sin 2 ( Ω 2 ) ⋅ cot ( Ω 2 ) ) 2

Compared with (2.8), the power of the sin() term is increased to 2nd power, which results in a higher steepness at the corner frequency (see Figure 4). The corner frequency is calculated from

(3.5) ( ω 3 4 ⋅ ( 1 − ω ) ( 2 − ω ) ⋅ cot ( ½ Ω 3 dB ) sin 2 ( ½ Ω 3 dB ) ) 2 = 1

Figure 4:

Transfer function of 1st–3rd order DC blocker filters.

For small corner frequencies the series expansion again finally results in

(3.6) ω ≈ Ω 3 dB

The filter coefficients compute as:

a 1 = + 6 − 7 ⋅ Ω 3 dB 2 − Ω 3 dB

a 2 = − 6 + Ω 3 dB 2 − Ω 3 dB ⋅ ( 1 − Ω 3 dB ) 2

a 3 = + ( 1 − Ω 3 dB ) 2

b 0 = + ( 1 − Ω 3 dB )

b 1 = − 3 ⋅ ( 1 − Ω 3 dB )

b 2 = + 3 ⋅ ( 1 − Ω 3 dB )

(3.7) b 3 = − ( 1 − Ω 3 dB )

Finally, we report without proof that for small Ω_3 dB the filter is stable. The pole’s absolute values are approximately at |z| ≈ (1 − ω).

Figure 4 shows the transfer functions 1st–3rd order for filters with corner frequencies Ω_3dB = 1/32 and Ω_3dB = 1/8. It is clearly visible that the steepness of the transition region between pass and stop regions increases with increasing filter order. Please notice the reduced x axis range when comparing with Figures 1 and 3.

4 Low-pass filters

The same principle can also be applied to low-pass filters with a corner frequency near to the Nyquist frequency. Also such filters are needed in communication systems, for example in those based on OFDM modulation [7], [8].

We sketch the calculations for the 1st order filter here. The starting point is again (1.3), but for a low-pass filter we have to swap the boundary conditions: H(Ω = 0) = 1 (frequency zero shall pass) and H(Ω = π) = 0 (Nyquist frequency shall be blocked). Equations (1.4) and (1.5) are the replaced by

(4.1) H ( Ω = 0 ) = b 0 + b 1 1 − a 1 = 1

(4.2) H ( Ω = π ) = b 0 − b 1 1 + a 1 = 0

or, in other words, b ₀ = b ₁ and b ₀ + b ₁ = 1 − a ₁. Combining leads to 2b ₀ = 2b ₁ = 1 − a ₁. We set ω = 1 + a ₁ and insert these relations into (6):

(4.3) | H ( Ω ) | 2 = ½ ( 2 − ω ) 2 ⋅ ( 1 + cos ( Ω ) ) 1 + ( ω − 1 ) 2 − 2 ⋅ ( ω − 1 ) ⋅ cos ( Ω ) = 1 1 + ( ω 2 − ω tan ( Ω 2 ) ) 2

Compared with (1.9), the cot() function is replaced with tan(). The corner frequency, where the filter’s attenuation reaches 3 dB, computes as:

| H ( Ω 3 dB ) | 2 = 1 2 = 1 1 + ( ω 2 − ω ⋅ tan ( Ω 3 dB 2 ) ) 2

( ω 2 − ω ⋅ tan ( Ω 3 dB 2 ) ) 2 = 1

(4.4) cot ( Ω 3 dB 2 ) = ± ω 2 − ω

The corner frequency is now near to the Nyquist frequency, which corresponds to Ω = π. This means (π − Ω_3dB) << 1. Under this condition, we transform (4.4) using the relation cot(x) = tan(π/2 − x). After that, the normal series expansion for tan(x) is possible.

cot ( Ω 3 dB 2 ) = tan ( π − Ω 3 dB 2 ) = ± ω 2 − ω

(4.5) π − Ω 3 dB ≈ ± 2 ⋅ ω 2 − ω ≈ ω

Our parameter ω equals the distance of the (normalized, angular) 3 dB corner frequency of the filter to the Nyquist frequency (half sampling rate). If the corner frequency is given, we can calculate the filter coefficients b ₀, b ₁ and a ₁ (see text above (4.3)):

a 1 = ( π − Ω 3 dB ) − 1

b 0 = 1 − ½ ( π − Ω 3 dB )

(4.6) b 1 = 1 − ½ ( π − Ω 3 dB )

The low-pass filter coefficients (4.6) are equivalent to their high-pass counterparts (1.12) except for the sign of a ₁ and b ₁. Inserting these coefficients into (1) results in the instruction to compute the output samples y _k of the filter from the input x _k.

(4.7) y k = 1 − ω 2 ⋅ x k + x k − 1 − 1 − ω ⋅ y k − 1

The filter equations and coefficients for higher-order low-pass filters can be derived in the same way as for the DC blocker filters. This is left to another publication.

5 Implementation aspects

5.1 Calculating filter coefficients

When practically using the above results for designing one of the investigated IIR high-pass filters, one first has to calculate the normalized angular corner frequency (filter attenuation 3 dB) that equals our filter parameter ω. From (4) we get:

(9) ω = Ω 3 dB = 2 π ⋅ f 3 dB f S

where f _3 dB is the corner frequency in user units (e.g., MHz) and f _S is the sampling rate of the filter in the same units.

The filter equation for the first order DC blocker (or high-pass) filter is then given by:

(10) y k = 1 − ω 2 ⋅ x k − x k − 1 + 1 − ω ⋅ y k − 1

For the second order filter it is given by:

(11) y k = ( 1 − ½ ω ) ⋅ ( x k − 2 x k − 1 + x k − 2 ) + ( 2 − 2 ⋅ ω − ½ ω 2 ) ⋅ y k − 1 − ( 1 − ½ ω ) 2 ⋅ y k − 2

For the third order filter this equation is given by:

(12) y k = ( 1 − ω ) ⋅ ( x k − 3 x k − 1 + 3 x k − 2 − x k − 3 ) + 6 − 7 ω 2 − ω ⋅ y k − 1 − 6 + ω 2 − ω ⋅ ( 1 − ω ) 2 ⋅ y k − 2 + ( 1 − ω ) 2 ⋅ y k − 3

In many cases, the exact value of the corner frequency is not so critical. Then, one can attempt to choose the parameter ω (or 1 / 2 ⋅ω in case of the 2nd order filter) to be equal to a negative power of 2, like 1/8, 1/32, or so. For 1st and 2nd order filters, all filter coefficients can be well expressed as simple fractions in this case. Thus, all mathematical operations can be well executed with integer arithmetics in hardware. This reduces the detrimental effects of rounding errors, which might even lead to instabilities.

5.2 VHDL simulation

To test the designed filter architectures, we have undertaken a simulation of example filters in time domain in the hardware description language VHDL, which would be a preferred candidate for a filter implementation in digital hardware. The simulation presented here uses real arithmetics for all data, but also fixed point simulations were carried out.

Figure 5 shows simulation waveforms of 1st–3rd order filters with ω = 1/8 for the frequency range 0 < Ω/π < 0.1 (see also Figure 4). The pink wave at the top is the input chirp signal (frequency varies from 0 to 0.1 f _N = 0.05 f _S). The three blue waveforms are the outputs of the 1st–3rd order DC blocker filters. The gray line above each approximates the un-squared transfer function |H(Ω)| (not |H(Ω)|²) as determined from the signal maxima. The red curve at the bottom gives the actual signal frequency as fraction of the Nyquist frequency.

Figure 5:

Simulated signal waveforms of 1st–3rd order DC blocker filters.

6 Conclusions

We have investigated a class of IIR filters, namely high-pass filters with very low corner frequency, also named as DC blockers. The proposed 2nd and 3rd order filters show a significantly smaller and steeper transition range between stop and pass-band than the commonly used 1st order filters. We have proven that the filters are intrinsically stable (i.e., when not considering rounding errors) and provide analytical formulas to calculate the filter coefficients from the desired cutoff frequency.

Corresponding author: Klaus Tittelbach-Helmrich, IHP – Leibniz-Institut für innovative Mikroelektronik, Im Technologiepark 25, D-15236 Frankfurt (Oder), Germany, E-mail: tittelbach@ihp-microelectronics.com

Funding source: European Horizon 2020 Programme

Award Identifier / Grant number: 857201

Acknowledgment

The author wishes to thank Nebojsa Maletic for critically reading and commenting the manuscript.

Author contribution: The author had accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: The research leading to these results has received funding by the European Horizon 2020 Programme for research, technological development and demonstration under Grant Agreement Number 857201 (H2020 5G-VICTORI).
Conflict of interest statement: The author declares no conflicts of interest regarding this article.

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Received: 2020-10-15

Accepted: 2021-01-18

Published Online: 2021-02-18

Published in Print: 2021-10-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/freq-2020-0177

Keywords for this article

DC blocker; digital filter; IIR filter

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BY 4.0