Logarithmic representation, frequency response, cut-off frequency

Logarithmic scales are often used in electrical engineering and communications engineering to represent the frequency response of filters and amplifiers. The logarithmic representation provides a better overview of the behaviour of the system across the entire frequency range.

Figure 1 shows two comic strips from XKCD that humorously illustrate the advantages of logarithmic representation.

In this chapter, we will look at the logarithmic representation of frequency responses in the form of so-called Bode diagrams. To describe the frequency response of simple filter circuits, the unit decibel, as well as the cut-off frequency and the order of filters, are defined. In addition, bandpass and band-stop filters are introduced.

1 Logarithmic representation of power functions

In logarithmic representation, a logarithmic scale is used for one or more axes. A linear increase in the values to be plotted corresponds to a logarithmic increase in the distance on the scale. Conversely, this can be illustrated more clearly as follows: With a linear increase in the distance on the scale, the power of the values to be plotted increases linearly by the same amount.

Logarithmic scales are generally useful for representing value ranges across several orders of magnitude. A special feature of the double logarithmic representation is that power functions are represented as straight lines. Figure 2 shows a comparison of several power functions in linear representation (left) and in double logarithmic representation with base \(10\) (right).

In double logarithmic representation, the exponent of a power function corresponds to the slope of the straight line. A shift in the y-direction corresponds to a pre-factor \(a\) for the function \(f(x)\), a shift in the x-direction corresponds to a pre-factor \(x_{\mathrm {off}}\) for x in the function. These properties make logarithmic representation particularly suitable for displaying amplitude responses.

2 Definition of decibel

Decibel is an auxiliary unit of measurement used to indicate the decimal logarithmic ratio of two quantities. It is used in signal theory and communications engineering, for example to indicate the amplification/attenuation of a component or a signal path. One decibel \([\mathrm {\mathrm{d}B }]\) corresponds to ten times the base unit bel \([\mathrm {B}]\).

The bel is defined as the designation of the decimal logarithmic ratio (symbol \(Q\)) of two power quantities (index \(P\)) with the same unit as shown in equation ??. In connection with voltage or current as so-called power root quantities (formerly field quantities, index \(F\)), the (deci-)bel can also be used in linear systems as shown in equation ??. The conversion is based on the proportionality of \(P\sim U^2\) and \(P\sim I^2\), respectively, which results in the factor \(2\).

\begin {align} &\textrm {Power indicators }(\textrm {e.g.}P,\ W)& Q_{\mathrm {(P)}} &= \mathrm {log}\frac {P_2}{P_1}\ \operatorname {B} = 10\cdot \mathrm {log}\frac {P_1}{P_2}\ \mathrm{d}B && \label {eq:def:dezibel:leistung}\\ &\textrm {Power root group}(\textrm {e.g.}U,\ I)& Q_{\mathrm {(R)}} &= \mathrm {log}\frac {U_2^2}{U_1^2}\ \operatorname {B} = 20\cdot \mathrm {log}\frac {U_1}{U_2}\ \mathrm{d}B && \label {eq:def:dezibel:wurzel} \end {align}

The value \(Q\) in \(\mathrm{d}B \) describes the amplification of a system from the input signal (index \(1\)) to the output signal (index \(2\)). Table 1 shows the corresponding voltage ratios \(\frac {U_2}{U_1}\) and power ratios \(\frac {P_2}{P_1}\) for typical decibel values. A change of \(6\mathrm{d}B \) corresponds, rounded to the second decimal place, exactly to the factor \(2\) for voltages and, rounded to the first decimal place, exactly to the factor \(4\) for powers.

Since \(3\mathrm{d}B \) corresponds very precisely to the factor \(2\) for power (factor \(\sqrt {2}\) for voltage) and \(6\mathrm{d}B \) to the factor \(4\), estimates can be made relatively easily. Compared to the SI unit neper, which is based on the natural logarithm, the decibel has therefore become established in practice as a logarithmic auxiliary unit of measurement.

Faustformeln für Spannungsverhältnisse:

• Factor \(1 = 0\mathrm{d}B \)
• Factor \(2 \approx 6\mathrm{d}B \)
• Factor \(4 \approx 12\mathrm{d}B \) with \(4 = 2^2 \approx 2\cdot 6\mathrm{d}B \)
• Factor \(5 \approx 14\mathrm{d}B \) with \(5 = 10 \cdot \frac {1}{2} \approx 20\mathrm{d}B - 6\mathrm{d}B \)
• Factor \(8 \approx 18\mathrm{d}B \) with \(8=2^3 \approx 3\cdot 6\mathrm{d}B \)
• Factor \(10 = 20\mathrm{d}B \) \(\Longrightarrow \) \(20 \mathrm{d}B / \mathrm {Dek}\) (voltage-related)

As an example, Figure 4 shows the logarithmic y-axis for \(A(\omega )\) once without units (left) and once in dB (right) with the integer factors or decibel values from the above rule of thumb labelled.

Phenologically, the following applies:

• There is amplification for \(\frac {U_2}{U_1}>1\)
• Attenuation occurs for \(\frac {U_2}{U_1}<1\)

For amplification and damping as physical quantities, the following applies: they correspond to each other in dB with opposite signs. The usual symbols are \(G\) (from ‘gain’) for the amplification of components and \(D\) (from ‘damping’) for their attenuation. Both values are typically given in \(\mathrm{d}B \) and, unless otherwise specified, in relation to voltage levels. In data sheets, attenuation is sometimes also given in \(\mathrm{d}B \) with a negative sign.

3 Cut-off frequency of simple filter circuits

High-pass and low-pass filters allow input signals to be filtered depending on their frequency, as shown in Chapter ?? for a low-pass filter and in Chapter ?? for a high-pass filter.

The frequency range of the amplitude responses of both filters is divided into a passband and a stopband. The so-called cutoff frequency \(f_g\) or limit (circuit) frequency \(\omega _g\) serves as the boundary. \(\omega _g\) is defined by a fixed ratio of effective output voltage to input voltage.

In the case of simple filter circuits, the following applies:

\begin {equation} \left .\frac {U_2}{U_1}\right \rvert _{\omega =\omega _g} := \frac {1}{\sqrt {2}} \leftrightarrow A(\omega _g) := \frac {1}{\sqrt {2}} \label {eq:def:grenzfrequenz} \end {equation}

According to Table 1, this corresponds to an attenuation of \(3\mathrm{d}B \) or an amplification of \(-3\mathrm{d}B \). The output power at \(\omega _g\) is half the input power.

Example: First-order low-pass filter

Using equation ??, the following applies to the first-order RC low-pass filter for the cut-off frequency: \begin {align} A(\omega ) &= \frac {1}{\sqrt {1 + (\omega CR)^2}} \nonumber \\ A(\omega _g) &= \frac {1}{\sqrt {1 + (\omega _g CR)^2}} \overset {!}{=} \frac {1}{\sqrt {2}} \nonumber \\ \mathrm{d}ots \Longrightarrow \omega _g &= \frac {1}{CR} \label {eq:tiefpass:grenzfrequenz} \end {align}

The cut-off frequency \(\omega _g\) corresponds to the reciprocal of the time constant \(\tau = CR\) of the RC element.

The frequency, amplitude and phase responses of the first-order RC low-pass filter from equations ??, ?? and ?? an be normalised to \(\omega _g\). This results in the following component-independent values: \begin {align} &&F(\mathrm {j}\omega ) &\overset {\text {\tiny {\ref {eq:tiefpass:frequenzgang}}}}{=} \frac {1}{1+\mathrm {j}\omega CR} & &\overset {\text {\tiny {\ref {eq:tiefpass:grenzfrequenz}}}}{\Longrightarrow }& F(\mathrm {j}\omega ) &= \frac {1}{1+\mathrm {j}\left (\omega /\omega _{\mathrm {g}}\right )} &&\label {eq:tiefpass:normiert:frequenzgang}\\ &&A(\omega ) &\overset {\text {\tiny {\ref {eq:tiefpass:ampli}}}}{=} \frac {1}{\sqrt {1 + (\omega CR)^2 }} & &\overset {\text {\tiny {\ref {eq:tiefpass:grenzfrequenz}}}}{\Longrightarrow }& A(\omega ) &= \frac {1}{\sqrt {1 + \left (\omega /\omega _{\mathrm {g}}\right )^2 }} \label {eq:tiefpass:normiert:amplitudengang}\\ &&\varphi (\omega ) &\overset {\text {\tiny {\ref {eq:tiefpass:phase}}}}{=} -\arctan \left (\omega CR\right ) & &\overset {\text {\tiny {\ref {eq:tiefpass:grenzfrequenz}}}}{\Longrightarrow }& \varphi (\omega ) &= -\arctan \left (\omega /\omega _{\mathrm {g}}\right ) \label {eq:tiefpass:normiert:phasengang} \end {align}

Figure 3 shows the amplitude response as a function of \(\omega /\omega _g\). In addition to the actual curve in red, the curve of an idealised low-pass filter with high slope is also shown in blue.

A change in the cut-off frequency results in a compression or stretching of the amplitude response along the frequency axis in a linear representation.

Example: First-order high-pass filter

Similarly, we can determine the cut-off frequency for the first-order RC high-pass filter:

\begin {equation} \begin {aligned} A(\omega _g) &= \frac {1}{\sqrt {1+ \left ( \frac {1}{\omega _g CR} \right )^2 }} \overset {!}{=} \frac {1}{\sqrt {2}} \\\mathrm{d}ots \Longrightarrow \omega _g &= \frac {1}{CR} \end {aligned} \end {equation}

The cut-off frequency of both filters is therefore identical for the same component values \(R\) and \(C\). The frequency response of the high-pass filter is analogous to that of the low-pass variant:

\begin {align} F(\mathrm {j}\omega ) &= \frac {1}{1-\mathrm {j}\left (\omega _{\mathrm {g}}/\omega \right )} \label {eq:hochpass:normiert:frequenzgang}\\ A(\omega ) &= \frac {1}{\sqrt {1 + \left (\omega _{\mathrm {g}}/\omega \right )^2 }}\\ \varphi (\omega ) &= \arctan \left (\omega _{\mathrm {g}}/\omega \right ) \label {eq:hochpass:normiert:amplitudengang} \end {align}

4 First-order low-pass filter

Frequency responses are typically represented in Bode diagrams. Bode diagrams consist of a double logarithmic representation of the amplitude response and a single logarithmic representation of the phase response. The x-axis for the (circular) frequency is logarithmically scaled in both representations. The y-axis is logarithmically scaled for the amplitude response and linearly scaled for the phase response.

Figure 4 shows the amplitude response from Eq. ?? of a first-order lowpass filter as a Bode diagram. The passband for \(\omega < \omega _g\) and the stopband for \(\omega > \omega _g\) are marked, as is the position for \(\omega =\omega _g\). The frequency specification on the x-axis has been normalised to the cut-off frequency and is therefore unitless.

The division of the frequency range into passband and stopband by the cut-off frequency is clearly visible. The amplitude response curve can be approximated in both ranges using asymptotes (straight lines) [blue dotted lines]: \begin {align*} \lim _{\omega \to 0} A(\omega ) &= 1 & &\Longrightarrow & A(\omega ) &\approx 1 & &\text {für}& \omega &< \omega _g \text {(Passband)} \\ \lim _{\omega \to \infty } A(\omega ) &= \frac {\omega _g}{\omega } & &\Longrightarrow & A(\omega ) &\approx \frac {\omega _g}{\omega } & &\text {für}& \omega &> \omega _g \text {(Stopband)} \end {align*}

The slope of the asymptote (straight line) in the cut-off range is \(-20 dB/dec\), due to the proportionality of \(A(\omega ) \sim \omega ^{-1}\). The approximation by both asymptotes deviates by a maximum of \(3 \mathrm{d}B \) for \(\omega = \omega _g\) from the actual curve. At the cut-off frequency, both asymptotes intersect, which is not generally true for filters. Both asymptotes (straight lines) and the point \(A(\omega _g) = 1/{1}{\sqrt {2}}\) were suitable for constructing a sketch. For the sketch, an arc of asymptote is drawn from \(A(\omega _g)= 1/{1}{\sqrt {2}}\) to asymptote in the transition range of factor five greater or less than the cut-off frequency (\(\frac {1}{5}\,\omega _g < \omega < 5\,\omega _g\)) .

A change in the cut-off frequency causes a shift along the frequency axis in the Bode diagram. This applies to both the amplitude and phase response, as the frequency axis is logarithmically scaled in both representations. Normalisation to the cut-off frequency is useful for representing the functional behaviour of the frequency response independently of specific component values.

Figure 5 shows the corresponding phase response as a Bode diagram, also with the frequency normalised to the cut-off frequency. The curve shape of the phase response resembles the curve shape of an arctangent in linear representation with corresponding displacement and compression.

The range of values of the phase response is determined by the two horizontal asymptotes. \(\varphi (\omega ) = 0\ ^\circ \) für \(\omega \rightarrow 0\) und \(\varphi (\omega ) = -90\ ^\circ \) für \(\omega \rightarrow \infty \) begrenzt. In this representation, the phase curve has point symmetry at the inflection point in \(\varphi (\omega _g) = -45\ ^\circ \) and no extrema.

Let \(\omega _n\) be the normalised frequency \(\omega /\omega _g\). The phase response can then be approximated in sections by the following three straight lines: \begin {equation} \varphi (\omega ) \approx \begin {cases}\begin {aligned}0\ &^\circ &&\text {für}& &\omega _n < 10^{-1} &&\text {Passband (without Cutoff frequency)}\\ -45\ &^\circ + \frac {-45\ ^\circ }{\mathrm {Dek}}\omega _n &&\text {für}& 10^{-1} \leq \ &\omega _n \leq 10 &&\text {transition band}\\ -90\ &^\circ &&\text {für}& 10 <\ &\omega _n &&\text {Stopband (without Transition band)} \end {aligned} \end {cases} \end {equation}

The maximum deviation of the approximation is \(\pm \ 5.71^\circ \) at the inflection points at \(\omega _g\pm 1\ \mathrm {dec}\).

4.1 Comparison of linear and logarithmic representations

For comparison, Figure 6 shows the amplitude and phase response of a first-order lowpass filter in linear and logarithmic representation (Bode diagram).

Unlike linear representation, in logarithmic representation the derivative of a function does not correspond to the readable slope in the curve. This can be clearly seen in Figure 6. The slope of \(A(\omega )\) in linear representation approaches \(0\) approaches \(0\) for \(\omega \to 0\) as well as for \(\omega \to \infty \). [See chapter ??] In the Bode diagram, the slope for \(\omega \to 0\) is also zero, but approaches approaches a minimum slope of \(-20\ \mathrm{d}B /\mathrm {dec}\) for \(\omega \to \infty \). In the phase response, the slope (derivative) for \(\omega \to 0\) is minimal and negative in linear representation. However, in logarithmic representation, \(\varphi (\omega )\) appears flattest (slope towards \(0\)) for \(\omega \to 0\).

4.2 Comparison of first-order low-pass and high-pass filters

Figure 7 shows the frequency responses of a lowpass and a highpass 1st order filter in a Bode diagram for comparison. The frequency range (x-axis) is normalised to the respective cut-off frequency in both cases.

5 Bandpass, Bandstop

Bandpass filters and band-stop filters are filter types used to filter frequency bands.

Similar to low-pass and high-pass filters, both are widely used in signal processing, for example in the fields of audio, communications, measurement and control technology. Other applications can be found, for example, in energy technology when feeding energy into the electrical supply network to ensure a stable mains frequency.

Bandpass filters filter out signals outside a specific frequency band, while band-stop filters filter out signals within a specific frequency band. The frequency bands are defined by an upper and lower cut-off frequency \(f_{\mathrm {go}}\) and \(f_{\mathrm {gu}}\).

\begin {equation} \left .\frac {\mathrm {U}_2}{\mathrm {U}_1}\right \rvert _{f=f_{\mathrm {go}}} = \frac {1}{\sqrt {2}} \text {und} \left .\frac {\mathrm {U}_2}{\mathrm {U}_1}\right \rvert _{f=f_{\mathrm {gu}}} = \frac {1}{\sqrt {2}} \end {equation}

Derived quantities are the bandwidth \(B\) and the centre frequency \(f_{\mathrm {M}}\). The bandwidth is defined as the difference between the upper and lower cut-off frequencies:

\begin {equation} \label {eq:def:b} B = \Delta f = f_{\mathrm {go}} - f_{\mathrm {gu}} \end {equation}

The centre frequency \(f_{\mathrm {M}}\) is defined as the geometric mean of both cut-off frequencies:

\begin {equation} \label {eq:def:fm} f_{\mathrm {M}} = \sqrt {f_{\mathrm {go}} \cdot f_{\mathrm {gu}}} \end {equation}

In logarithmic representation, \(f_{\mathrm {M}}\) is, by definition, exactly in the middle between both cut-off frequencies. This allows the size and position of the frequency band in the frequency spectrum to be described using \(B\) and \(f_{\mathrm {M}}\).

5.1 Implementation: Series or parallel connection of low-pass and high-pass filters

Bandpass and band-stop filters can be implemented by combining a low-pass and high-pass filter. This provides a clear illustration of how combined filter circuits behave.

Connecting a low-pass and high-pass filter in series creates a band-pass filter, provided that the cut-off frequencies are selected appropriately. For band-pass behaviour, the upper cut-off frequency must be realised by the low-pass filter and the lower cut-off frequency by the high-pass filter. Mathematically, the resulting frequency response can be described as the product of the individual frequency responses with \(\underline {F}_{\mathrm {BP}} = \underline {F}_{\mathrm {TP}} \cdot \underline {F}_{\mathrm {HP}}\). A prerequisite for multiplying the individual frequency responses to obtain the overall frequency response is that the downstream four-pole circuit has no feedback on the output signal of the upstream four-pole circuit.

Figure 8 shows an example of the circuit diagram of such a second-order bandpass as a series connection of an RC low-pass filter and an RC high-pass filter, both of the first order. The symbolic amplifier block between the two filters prevents the high-pass filter from having a feedback effect on the output voltage of the low-pass filter. Only under the assumption of no feedback can the frequency responses of the low-pass and high-pass filters be multiplied by connecting them in series.

A band-stop filter can be implemented by connecting the inputs of the low-pass and high-pass filters in parallel and their outputs in series. This ensures that both individual filters receive the same input voltage and the output voltages are added together. The resulting frequency response is the sum of the individual frequency responses with \(\underline {F}_{\mathrm {BS}} = \underline {F}_{\mathrm {TP}} + \underline {F}_{\mathrm {HP}}\). In this case, the upper cut-off frequency must be implemented by the high-pass filter and the lower cut-off frequency by the low-pass filter.

5.2 Bode diagram

Figure 9 shows the Bode diagram of the bandpass from Fig. 8. The amplitude response and phase response are superimposed on a common X-axis for better comparison of the two graphs. The gain factor is \(1\) (passive) and the cut-off frequencies are \(10^2\mathrm {Hz}\) and \(10^4\mathrm {Hz}\).

The upper and lower cut-off frequencies \(\omega _{\mathrm {go}}\) and \(\omega _{\mathrm {gu}}\) are marked in both representations (red dashed line, vertical), as is the \(-3\ \mathrm{d}B \) limit (black dashed line, horizontal) whose intersection with \(A(\omega )\) defines the cut-off frequencies.

The bandwidth is indicated in the amplitude response as a double arrow between the two cut-off frequencies. Due to the logarithmic scaling, the bandwidth does not correspond to the geometric distance shown in the diagram.

The diagram shows: \begin {align} \omega _{\mathrm {gu}} &= 10^2\mathrm {Hz} \qquad \text {und} \qquad \omega _{\mathrm {go}} = 10^4\mathrm {Hz}\vphantom {\bigg |}\\ B &\overset {\tiny \ref {eq:def:b}}{=} \Delta f = \frac {\omega _{\mathrm {go}} - \omega _{\mathrm {gu}}}{2\pi } = \frac {9900}{2\pi }\mathrm {Hz}\vphantom {\bigg |}\\ f_{\mathrm {m}} &\overset {\tiny \ref {eq:def:fm}}{=} \frac {\omega _{\mathrm {m}}}{2\pi } = \frac {\sqrt {\omega _{\mathrm {go}} \cdot \omega _{\mathrm {gu}}}}{2\pi } = \frac {10^3}{2\pi }\mathrm {Hz}\vphantom {\bigg |} \end {align}

The centre frequency \(\omega _{\mathrm {m}}\) is marked in the phase response. The phase shift there is \(0^\circ \), since the phase shifts of the low-pass and high-pass (blue dashed line, thin) cancel each other out at this point.

The high-pass behaviour in the lower cut-off range (with \(+20\mathrm{d}B /\mathrm {Dek}\) and \(\varphi (\omega ) > 0^\circ \)) and the low-pass behaviour in the upper cut-off range (with \(-20\mathrm{d}B /\mathrm {Dek}\) and \(\varphi (\omega ) > 0^\circ \)) are clearly visible.

6 Order of filters, example filter of higher order

Figure 10 shows an example comparison of the amplitude responses of various higher-order filters. A Butterworth, Chebyshev, and Cauer filter are shown in comparison with an ideal low-pass filter.

Explain the order based on the total steepness of the flanks.

• Low-pass/high-pass (one edge): Order \(n\) at \(\pm \ n \cdot 20\mathrm{d}B /\mathrm {dec}\)
• Band-pass/band-stop (two edges): Order \(2n\) at \(\pm \ n \cdot 20\mathrm{d}B /\mathrm {dec}\)

Second-order filter through: RLC or combination of two first-order filters: 2x (RC) or 2x (RL) (first order)

Filters of higher order than 2 are possible by cascading lower-order filters.

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