8.3.1 Level Measurement

Level measurements [McN84] can be performed with the systems shown in Figs. 8.5 and 8.9. For PEAK measurement, the absolute value of the input is compared with the peak value . If the absolute value is greater than the peak value, the difference is weighted with the coefficient AT (attack time) and added to . For this attack case , we get the difference equation (see Fig. 8.5)

(8.12)

with the transfer function

(8.13)

If the absolute value of the input is smaller than the peak value (the release case), the new peak value is given by

(8.14)

with the release time coefficient RT. The difference signal of the input will be muted by the nonlinearity such that the difference equation for the peak value is given according to Eq. (8.14). For the release case, the transfer function

(8.15)

is valid. For the attack case, the transfer function (8.13) with coefficient AT is used, and for the release case, the transfer function (8.15) with the coefficient RT is used. The coefficients (see Section 8.3.3) are given by

(8.16)

(8.17)

where the attack time and the release time are given in msec ( sampling interval). With this switching between filter structures, one achieves fast attack responses for increasing input signal amplitudes and slow decay responses for decreasing input signal amplitudes.

A first variation for peak detection is the branched peak detector that has two different branches of operation, as shown in Fig. 8.6. For the attack case AC when and the release case RC when , the operations are given by

(8.18)

One further variation of the peak detector is the so‐called decoupled peak level detector, which is presented in Fig. 8.7. It computes an auxiliary signal based on the release time and then feeds this signal to an attack filter, as given by

(8.19)

(8.20)

This detector adds a small onset to the release time, as shown in Fig. 8.8. This leads to the release time approximately equal to and is therefore always bigger than the attack time. This can be useful to avoid artifacts arising from release times that are too short [Gia12].

The computation of the RMS value

(8.21)

over input samples can be achieved by a recursive formulation. The RMS measurement shown in Fig. 8.9 uses the square of the input and performs averaging with a first‐order lowpass filter. The averaging coefficient

(8.22)

is determined according to the time constant calculation discussed in Section 8.3.3, where is the averaging time in msec. The difference equation is given by

(8.23)

with the transfer function

(8.24)

8.3.2 Gain Factor Smoothing

Attack and release times can be implemented by the system shown in Fig. 8.10 [McN84]. The attack coefficient AT or release coefficient RT is obtained by comparing the input control factor with the previous one. A small hysteresis curve determines whether the control factor is in the attack or release status and hence gives the coefficient AT or RT. The system also serves to smooth the control signal. The difference equation is given by

(8.25)

with or and the corresponding transfer function leads to

(8.26)

8.3.3 Time Constants

If the step response of a continuous‐time system is

(8.27)

then sampling (step‐invariant transform) the step response gives the discrete‐time step response

(8.28)

The Z‐transform leads to

(8.29)

With the definition of attack time , we derive

(8.30)

(8.31)

The relationship between attack time and the time constant of the step response is obtained as follows:

(8.32)

Hence, the pole is calculated as

(8.33)

A system for implementing the given step response is obtained by the relationship between the Z‐transform of the impulse response and the Z‐transform of the step response:

(8.34)

The transfer function can now be written as

(8.35)

with the pole adjusting the attack, release, or averaging time. For the coefficients of the corresponding time constant filters, the attack case is given by Eq. (8.16), the release case by Eq. (8.17), and the averaging case by Eq. (8.22). Figure 8.11 shows an example where the dotted lines mark the and times.

8.4.1 Limiter

The block diagram of a limiter is presented in Fig. 8.12. The signal is determined from the input with variable attack and release times. The logarithm to the base 2 of this peak signal is taken and compared with the limiter threshold. If the signal is above the threshold, the difference is multiplied by the negative slope of the limiter LS. After this, the antilogarithm of the result is taken. The obtained control factor is then smoothed with a first‐order lowpass filter (SMOOTH). If the signal lies below the limiter threshold, the signal is set to . The delayed input is multiplied by the smoothed control factor to give the output .

8.4.2 Compressor

Feedback implementation

A DRC system can be implemented in a feedback form, as shown in Fig. 8.13, where level sensing is performed on the output signal . A delay has to be introduced in the feedback side‐chain path. The gain is then applied to the input signal .

Ducking

Ducking consists of using a second signal as input to the side‐chain of the DRC system, as shown in Fig. 8.14. This has various applications, for example, in the case of background music playing and an announcement has to be made. The level of the music can be automatically reduced by sensing the level of the speaker and thus applying a negative gain to the music (compression). This effect is also widely used in modern music production to give a stronger feeling of energy to the kick drum. In this case, the kick drum is used as a side‐chain signal to periodically attenuate the rest of the instruments.

Lookahead

The lookahead consists of introducing a delay in the direct path to sense the level prior to applying it, as shown in Fig. 8.15. It is useful when applied on transients or fast changing signals, because it predicts the variations and reduces (or avoids) the time needed to react to changes in the signal dynamic. When realized offline, the delay introduced can be compensated, but in real‐time, the delay is equal to the lookahead time. Given a lookahead time in ms, the number of samples is given by .

8.4.3 Compressor, Expander, Noise Gate

The block diagram of a compressor/expander/noise gate is shown in Fig. 8.16. The basic structure is similar to the limiter. In contrast to the limiter, the logarithm of the signal is taken and multiplied by 0.5. The obtained value is compared with three thresholds to determine the operating range of the static curve. If one of the three thresholds is crossed, the resulting difference is multiplied by the corresponding slope (CS, ES, NS) and the antilogarithm of the result is taken. A following first‐order lowpass filter provides the attack and release times.

8.4.4 Combination System

A combination of a limiter that uses PEAK measurement and a compressor/expander/noise gate that is based on RMS measurement, is presented in Fig. 8.17. The PEAK and RMS values are measured simultaneously. If the linear threshold of the limiter is crossed, the logarithm of the peak signal is taken and the upper path of the limiter is used to calculate the characteristic curve. If the limiter threshold is not crossed, the logarithm of the RMS value is taken and one of the three lower paths is used. The additive terms in the limiter and noise gate paths result from the static curve. After going through the range detector, the antilogarithm is taken. The sequence is smoothed with a SMOOTH filter in the limiter case, or weighted with corresponding attack and release times of the relevant operating range (compressor, expander, or noise gate). By limiting the maximum level, the dynamic range is reduced. As a consequence, the overall static curve can be shifted up by a gain factor. Figure 8.18 demonstrates this with a gain factor equal to 10 dB. This static parameter value is directly included in the control factor .

As an example, Fig. 8.19 illustrates the input , the output , and the control factor of a compressor/expander system. It is observed that signals with high amplitude are compressed and those with low amplitude are expanded. An additional gain of 12 dB shows the maximum value of 4 for the control factor . The compressor/expander system operates in the linear region of the static curve if the control factor is equal to 4. If the control factor is between 1 and 4, the system operates as a compressor. For control factors lower than 1, the system works as an expander ( and ). The compressor is responsible for increasing the loudness of the signal whereas the expander increases the dynamic range for signals of small amplitude.

8.5.1 Sampling Rate Reduction

To reduce the computational complexity, downsampling can be carried out after calculating the PEAK/RMS value (see Fig. 8.20). As the signals and are already band limited, they can be directly downsampled by taking every second or fourth value of the sequence. This downsampled signal is then processed by taking its logarithm, calculating the static curve, taking the antilogarithm, and filtering with corresponding attack and release time with reduced sampling rate. A following upsampling by a factor of 4 is achieved by repeating the output value four times. This procedure is equivalent to upsampling by a factor 4 followed by a sample‐and‐hold transfer function.

The nesting and spreading of partial program modules over four sampling periods is shown in Fig. 8.21. The modules PEAK/RMS (i.e. PEAK/RMS calculation) and MULT (delay of input and multiplication with ) are performed every input sampling period. The number of processor cycles for PEAK/RMS and MULT are denoted by Z1 and Z3, respectively. The modules LD(x), CURVE, , and SMO have a maximum number of processor cycles of Z2 and are processed consecutively in the given order. This procedure is repeated every four sampling periods. The total number of processor cycles per sampling period for the complete dynamics algorithm results from the sum of all three modules.

8.5.2 Curve Approximation

In addition to taking the logarithm and antilogarithm, other simple operations like comparisons and addition/multiplication occur in calculating the static curve. The logarithm of the PEAK/RMS value is taken as follows:

(8.36)

(8.37)

First, the mantissa is normalized and the exponent is determined. The function is then calculated by a series expansion. The exponent is simply added to the result.

The logarithmic weighting factor and the antilogarithm are given by

(8.38)

(8.39)

Here, is a natural number and is a fractional number. The antilogarithm is calculated by expanding the function in a series and multiplication by . A reduction of computational complexity can be achieved by directly using tables for log and antilog.

8.5.3 Stereo Processing

For stereo processing, a common control factor is needed. If different control factors are used for both channels, limiting or compressing one of the two stereo signals causes a displacement of the stereo balance. Figure 8.22 shows a stereo dynamic system in which the sum of the two signals is used to calculate a common control factor . The following processing steps of measuring the PEAK/RMS value, downsampling, taking logarithm, calculating static curve, taking antilogarithm attack and release time, and upsampling with a sample‐and‐hold function remain the same. The delay (DEL) in the direct path must be the same for both channels.

8.8.1 Introduction

Source‐filter separation and processing [Arf11] has been extensively used to extract the so‐called spectral envelope (filter coefficients) and the source signal from an audio signal. When applying the spectral envelope to the source signal again, the filtering operation will perfectly reconstruct the original. In the case of a speech signal, the spectral envelope has the advantage to provide a good representation of its formants. The idea of using the extracted formants to re‐synthesize voice has been expressed for a long time in [Sla62] and is nowadays used in applications such as Vocoders [Nag09] or artificial speech models [Kad07].

The system proposed here will apply some dynamic processing on the error signal (source signal) to affect the re‐synthesized result. However, too drastic modifications of the source signal can generate artifacts at the re‐synthesis stage. One of the applications presented here aims to reduce background noise and is especially effective when applied to speech recordings.

Source‐filter separation [Arf11], as shown in Fig. 8.28, uses linear predictive coding and decoding (LPC) by generating a prediction of the input signal with an adaptive FIR filter of order . This filter estimates a spectral envelope of the input . The difference between the predicted signal and the input is called the prediction error and represents the source signal. The prediction error can be seen as a whitened version of . For speech signals, the prediction error (source signal) represents an excitation signal similar to the sound emitted by the vocal cords, which will be filtered by the vocal tract to give the spectral shape of the speech signal. Further dynamic range processing of the prediction error will lead to a processed source/error , which is then fed to the LPC decoding as shown in Fig. 8.28 to reconstruct the processed output .

The prediction filter of order used in the analysis and synthesis is updated at every new incoming sample using a buffer of the past samples from . The filter coefficients are updated according to the least‐mean‐square (LMS) method given by

(8.40)

for , where is the filter coefficient index and corresponds to the step size of the adaption. The predicted signal is calculated using

(8.41)

which represents FIR filtering of the input with the previously estimated filter coefficients. The transfer function for coding (analysis) is given by

(8.42)

The decoder for the re‐synthesis uses the inverse transfer function of the coder given by

(8.43)

This filter uses the coefficients calculated by the prediction filter and performs as an all‐pole filter. From an implementation point of view, this is simply using the filter in a feedback loop, as depicted in Fig. 8.28.

8.8.2 Combination with DRC

In this particular system, the error signal will be processed by a DRC system to influence the re‐synthesized signal , as depicted in Fig. 8.29. DRC produces generally a quite transparent effect. Therefore, the processed error signal will not be affected too drastically, so the reconstructed signal will be less prompt to artifacts arising from imperfect reconstruction.

8.8.3 Applications

Denoising

Dynamic range control has been one of the first tools used to denoise audio signals. As an example, a noise gate or expander attenuates the input signal when its level goes below a defined threshold. By setting this threshold right above the background noise level, the parts where no signal of interest is present will see their level reduced, thus denoised. This denoising method is, by definition, effective only when the energy of the input signal is low in the case of a speech signal when the speaker is not speaking or pausing between words.

In this denoising application, an expander is used as a DRC block, because it tends to be less prompt to artifacts than a noise gate. The expander chosen in this application is composed of a peak level detector [Gia12]. Because the error signal carries out most of the background noise, the expander threshold parameter can be set to a few dBs above the noise level (see Fig. 8.30). The pulse train remains, whereas the noise in between pulses is slightly reduced. Moreover, the silent part sees its background noise reduced as well, because the level is located below the expander threshold. For this application, the time constants of the expander have been set to a very fast attack  ms to leave the transients and pulses untouched and a slower release  ms to avoid artifacts. A ratio has been chosen to drastically reduce the quieter sounds, thus the background noise. In this particular case, the threshold has been set up manually using prior knowledge of the noise level. However, an adaptive threshold may be applied for non‐stationary noise situations. It is worth mentioning that for high noise levels (that cover the speech), the presented algorithm will see its performances reduced drastically because its fundamental principle is to increase the level differences between parts of the signal. As mentioned in [Arf11], LPC can be parameterized so that the error signal resembles a whispering effect. This means that the resonant part of the voice is removed resulting in a combination of plosives, noise and pulse sequences. Removing the background noise from such a signal results in denoising the re‐synthesized signal. To obtain an excitation signal with such properties, filter orders in the range are most suitable (with  kHz).

The top left plot in Fig. 8.31 shows the spectrogram of a female singing voice with white noise (30 dB SNR). The top right plot spectrogram in Fig. 8.31 shows the effect of an expander applied as denoiser. In the case of the presence of stationary low‐colored background noise in a signal, it will be contained in the error signal as well, as the separation by the LPC is whitening the signal. By denoising the error signal with an expander, the re‐synthesized signal will also be denoised in parts where voicing is happening. This effect is visible on the spectrogram shown in the lower plot of Fig. 8.31, where an attenuation of the background noise is occurring without altering the original signal.

Transient Control

The error signal obtained by the source–filter separation contains the transients of the original sound. Because transients are sudden burst of energy, the prediction filter is not fast enough to predict them. This arises from to the weighted update of the filter coefficients (). Moreover, transients are generally broadband. This character makes them prompt not to be modeled by the filter, and therefore are present in the error signal.

Once transients are extracted, it is possible to choose the amount to re‐inject into the error signal before the re‐synthesis. For this matter, a gain controlling the amount of transient to add or cancel (by phase inversion) is placed, as shown in Fig. 8.32. One of the problems occurring frequently when recording speech is the presence of too strong transients or clipping. This can arise from the air flow hitting the microphone membrane when pronouncing a plosive. Reducing transients may be used to reduce this effect (pop filtering). Traditionally, compressors are used to overcome such problems. Nevertheless, it is also considered that the attack portion of sounds conveys brightness character, so increasing the transients may also give a brighter effect to the re‐synthesized signal. Depending on the desired effect, being able to control the amount of transients in a sound is useful. This property makes the error a good pre‐processing step to extract transients. Processing it with a compressor can achieve a complete transient extraction.

An example of extracted transients is shown in Fig. 8.33 and the corresponding spectrograms are visible in Fig. 8.34. Selecting a slower attack time will allow for the transient portion of the sound to pass through before the compressor starts clamping. However, if the attack time is too slow, then too big a portion of the transient may pass through. A noise gate is cascaded after the expander to fully remove the non‐transients parts (see Fig. 8.32).

1. Lowpass Filtering for Envelope Detection

Generally, envelope computation is performed by lowpass filtering the input signal's absolute value or its square.

1. Sketch the block diagram of a recursive first‐order lowpass .
2. Sketch its step response. What characteristic measure of the envelope detector can be derived from the step response and how?
3. Typically, the lowpass filter is modified to use a non‐constant filter coefficient . How does depend on the signal? Sketch the response to a rect‐signal of the lowpass filter thus modified.

2. Discrete‐time Specialties of Envelope Detection

Taking absolute value or squaring are nonlinear operations. Hence care must be taken when using them in discrete‐time systems as they introduce harmonics the frequency of which may violate the Nyquist bound. This can lead to unexpected results, as a simple example shall illustrate. Consider the input signal .

1. Sketch , and for different values of .
2. Determine the value of the envelope after perfect lowpass filtering, i.e. averaging, . Note: As the input signal is periodical, it is sufficient to consider one period, e.g.
   
3. Similarly, determine the value of the envelope after averaging .

3. Dynamic Range Processors

Sketch the characteristic curves mapping input level to output level and input level to gain for, and describe briefly the application of:

1. limiter;
2. compressor;
3. expander; and
4. noise gate.