Digital DC blocker filters

3.1 First order

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

This corresponds to the filter equation

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

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:

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

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

We insert this into (6):

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.

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

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):

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.

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)):

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:

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

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

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)

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:

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

From the second condition we get:

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

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

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.

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:

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.

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):

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.

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:

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:

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

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

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:

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)

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:

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:

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

Figure 4:

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

For small corner frequencies the series expansion again finally results in

The filter coefficients compute as:

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.

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:

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:

For the second order filter it is given by:

For the third order filter this equation is given by:

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.