Acoustic feedback suppression in audio mixer for PA applications
Mattias Ekström
Department of Physics Linnaeus väg 20
901 87 Umeå
Sweden
Acoustic feedback suppression in audio mixer for PA applications
Mattias Ekström (maek0025@ student. umu. se ) June 19, 2017
Master’s thesis, engineering physics, spring 2017, 30 credits
Supervisor: Christian Schüld, Limes Audio
Abstract
When a speaker is addressing an audience, a PA system consisting of a microphone
and a loudspeaker is often used. If the microphone picks up too much of the loud-
speaker energy, acoustic feedback in the form of an unwanted characteristic howling
can occur. Limes Audio is a software company that specializes in improving sound
quality in digital communications, mainly conference telephony, and has developed
a reference product, the Magneto mixer, to demonstrate the capability of their soft-
ware TrueVoice. The company now wishes to expand the field of usage for the
Magneto mixer to enable it to work as a microphone mixer in PA scenarios, and for
this, a feedback suppression feature is needed. This master’s thesis aims at survey-
ing the market and the literature in the field and specifying the requirements for
a feedback suppression feature. Three methods for suppressing howling feedback
are evaluated through simulations and compared in terms of maximum stable gain
(MSG) and subjective listening experience. The method that performed the best
based on these criteria was acoustic feedback cancellation with a 5 Hz frequency
shift on the loudspeaker signal. This method makes use of an adaptive filter to
model the acoustic feedback path and to remove the feedback component from the
microphone signal. In the simulations, the method was able to increase the stable
gain by approximately 10 dB while maintaining a good sound quality.
Sammanfattning
När en talare talar för en publik används ofta ett PA system bestående av en mikro-
fon och en högtalare. Om mikrofonen tar upp för mycket av ljudet från högtalaren
finns en överhängande risk för akustisk rundgång i form av ett karaktäristiskt oöns-
kat tjut. Limes Audio är ett företag som utvecklar mjukvara för att förbättra ljud-
kvaliten i digital kommunikation, främst inom konferenstelefoni. De har utvecklat en
demonstrationsprodukt, Magnetomixern, som kan användas som en konferenstele-
fon för att demonstrera deras programvara TrueVoice. Företaget önskar nu utveckla
Magnetomixern till att även fungera som en ljudmixer för PA-scenarion, eller kon-
ferenstelefoni där intern ljudförstärkning i rummet behövs, och för detta behövs en
funktion för att ta bort eventuell rundgång. Detta examensarbete har som mål att
lägga grunden för en sådan funktion i Magnetomixern genom att undersöka markna-
den och litteraturen på området. Tre metoder för att eliminera rundgång utvärderas
i simuleringar och jämförs beträffande maximal stabil förstärkning (MSG) och sub-
jektiv ljudkvalitet. Metoden ”Acoustic feedback cancellation” tillsammans med ett 5
Hz frekvensskifte på högtalarsignalen gav högst MSG och bäst ljudkvalitet. Metoden
använder ett adaptivt filter för att approximera den akustiska återkopplingsvägen
mellan högtalare och mikrofon samt tar bort rundgångskomponenter från mikrofon-
signalen. I simuleringarna kunde metoden öka den maximala stabila förstärkningen
med upp till 10 dB medan en god ljudkvalitet på talet bibehölls.
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List of abbreviations
AEC Acoustic Echo Cancellation AEQ Automatic Equalization
AFC Acoustic Feedback Cancellation FFT Fast Fourier Transform
FIR Finite Impulse Response IIR Infinite Impulse Response
IMSD Interframe Magnitude Slope Deviation LTI Linear Time-Invariant
MSG Maximum Stable Gain
NFS Notch filter based Feedback Suppression NLMS Normalized Least Mean Square PA Public Address
PHPR Peak-to-Harmonic Power Ratio
PNPR Peak-to-Neighouring Power Ratio
RIR Room Impulse Response
1 Introduction 1
1.1 Background . . . . 1
1.2 Motivation . . . . 2
1.3 Objective . . . . 2
1.4 Disposition . . . . 2
2 Theory 4 2.1 Basics of signals and systems . . . . 4
2.1.1 Linear systems . . . . 4
2.1.2 Digital filters . . . . 6
2.2 The feedback phenomenon . . . . 6
2.3 Stability analysis . . . . 9
3 Methods used in feedback suppression 12 4 Description of algorithms 16 4.1 Frequency shifting . . . 16
4.1.1 Analytic signal . . . 17
4.2 Two-stage notch filtering . . . 18
4.2.1 Detection stage . . . 19
4.2.2 Suppression stage . . . 21
4.3 Acoustic feedback cancellation . . . 22
4.3.1 NLMS . . . 23
5 Method for testing 25 5.1 MATLAB simulation and evaluation . . . 25
6 Results 28
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6.1 Feedback suppression . . . 28
6.2 Maximum stable gain . . . 32
6.2.1 Frequency shifting . . . 32
6.2.2 Notch filters . . . 32
6.2.3 Acoustic feedback suppression . . . 33
6.3 Subjective listening experience . . . 34
7 Discussion, conclusion and future work 36
References 38
1 Introduction
1.1 Background
In any given situation where a speaker is addressing an audience using a Public Address (PA) system, consisting of a microphone and a loudspeaker, the entire performance is at risk of being ruined by feedback, perceived as ”howling” at a certain frequency. Feed- back howling is not only an unpleasant experience for the audience, but also puts the PA equipment at risk of being damaged. Feedback occurs when the microphone takes up too much of the loudspeaker’s energy (see chapter 2), and causes unstable oscillations at problematic frequencies which is perceived as howling, that probably is familiar to the reader. Throughout the history of PA systems, feedback has been a reoccurring phenomenon and different measures have been taken to prevent this unpleasant experi- ence. Since the 1960s, when the first feedback suppression methods were presented[1], [2], novel methods and algorithms have been proposed, and since the dramatic increase in the use of digital computers in the 1980s and forward, more powerful and efficient algorithms have been developed through software implementations in digital signal pro- cessors (DSP). Today, many consider the best method to avoid howling feedback to be a careful and well planned setup of the microphone and loudspeakers, along with an ex- perienced sound technician that sets the equalization in the PA system to be optimized for the specific room, and decrease the gain of potentially problematic frequencies [3].
In many applications though, there is a need for a plug-and-play solution without the
presence of a sound technician, and for these scenarios, the processes usually performed
by a sound technician must be automated or other measures needs to be taken in order
to avoid howling feedback.
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1.2 Motivation
Limes Audio AB is a company owned by Google that develops audio solutions for en- terprise applications. Their main product, TrueVoice, has been developed to remove echoes, noise and other sonic artefacts in conference telephony and other applications that makes use of a communication system with a loudspeaker and microphone situated in the same unit. Limes Audio has designed a reference product called the Magneto mixer, that can be used as a plug-and-play conference mixing unit together with a com- puter, and has the TrueVoice software embedded. The company now wishes to look into the possibility of expanding the field of usage for the Magneto mixer, from working as a conference telephony mixing unit to also be able to work as a plug-and-play mixer unit in a PA system, and other teleconferencing scenarios where internal sound reinforcement is necessary. For this, the software in the Magneto mixer needs to be adapted for the PA case, which has a different problem formulation than the teleconference case.
1.3 Objective
For the Magneto mixer to work properly in the PA case, there is a need for a feedback suppression feature. There are two main objectives for this work. The first objective is to survey the literature on the subject as well as the competitors solutions to the feedback problem, and provide documentation on the findings. The second objective is to specify the requirements for a feedback suppression feature in the Magneto mixer and to develop MATLAB code demonstrating the performance of some chosen methods, and to perform an evaluation regarding which method Limes Audio should aim at including in the Magneto Mixer in their future work of integrating a feedback suppressor in the Magneto mixer.
1.4 Disposition
Chapter 2 describes the mathematical theory of the feedback problem and the conditions required for howling feedback to occur. Chapter 3 briefly describes the available methods on the subject, and provides arguments for my choice of methods for the next section.
Chapter 4 describes the chosen feedback suppression algorithms in detail, and chapter 5
describes the methods used for testing the implementations and simulating the PA-setup
in MATLAB. Chapter 6 presents the results from the evaluation procedures and chapter 7
concludes the report with a discussion of the findings in the work, and suggestions for
future work.
2 Theory
This chapter describes the theoretical foundation upon which all feedback suppression algorithms are based, starting from the fundamentals in signals and systems. The math- ematical formulation of the feedback problem is presented, and the conditions required for howling feedback to occur are explained.
2.1 Basics of signals and systems
2.1.1 Linear systems
A system H is an operator that takes an input x(t) and produces an output y(t):
y(t) = H{x(t)}. (2.1)
H is said to be linear if it satisfies the superposition principle: if several inputs x 1 (t), x 2 (t), ..., x i (t) produces outputs
y 1 (t) = H{x 1 (t)} (2.2)
y 2 (t) = H{x 2 (t)} (2.3)
.. . (2.4)
y i (t) = H{x i (t)}, (2.5)
then the output upon addition of the inputs and possibly scaling them by factors α i
satisfies
α 1 y 1 (t) + ... + α i y i (t) = H{α 1 x 1 (t) + ... + α i x i (t)}. (2.6) A system is furthermore said to be time-invariant if a time shift T in the input only results in a corresponding time shift in the output:
y(t − T ) = H{x(t − T )}. (2.7)
A Linear Time-Invariant (LTI) system can be described by its impulse response h(t) in the time domain and by its frequency response H(ω) in the frequency domain. The impulse response is the output from an LTI system being excited with an impulse at time t = 0. In the discrete domain, this impulse is represented by the Kronecker delta impulse
d i =
0 if i 6= 0
1 if i = 0. (2.8)
The corresponding impulse in the continuous domain is the Dirac delta function. If the impulse response is known, one can, for any input x(t), determine the output y(t) of the system with the convolution operator ∗:
y(t) = h(t) ∗ x(t). (2.9)
The frequency response, H(ω) is obtained by computing the Fourier transform of the impulse response h(t), and describes the frequency spectrum of the output of the LTI system, when the input is one of the above described impulse functions:
H(ω) = F{h(t)}, (2.10)
where F is the Fourier transform operator. A property of interest for the convolution operator is the convolution theorem, which states that, upon computing the Fourier transform of both sides of eq. (2.9):
Y (ω) = F{h(t) ∗ x(t)} = F{h(t)}F{x(t)} = H(ω)X(ω), (2.11)
where Y (ω), X(ω) are the Fourier transforms of their corresponding signal [4].
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From eq. (2.11), it can easily be deduced that the total frequency response of a system can be found by dividing the Fourier transform of the output signal by the Fourier transform of the input signal:
H(ω) = Y (ω)
X(ω) . (2.12)
For real-valued signals, the corresponding Fourier transforms are complex and Hermi- tian [4]. From the complex-valued frequency response H(ω), the magnitude response
|H(ω)| and the phase response ∠H(ω) can be computed. These quantities describe the magnitude and phase of the frequency components in the output signal from the system.
2.1.2 Digital filters
A digital filter is a system that manipulates an input signal in a desired way to produce a specific output. Examples of these are band pass filters, low pass filters and high pass filters. Digital filters can be either Finite Impulse Response (FIR), or Infinite Impulse Response (IIR). As the names suggests, the impulse response of a FIR filter is of finite order, and infinite for an IIR filter. Since FIR filters have finite impulse responses, they are always stable, but can be computationally demanding, as opposed to IIR filters, that can sometimes be unstable, but are in general less computationally demanding than FIR filters [4].
2.2 The feedback phenomenon
In situations where a speaker is addressing an audience located in the same room, a PA system, consisting of a microphone and loudspeakers, is often used. Due to the fact that the microphone and loudspeaker are situated in the same room, there is a significant risk of feedback from the loudspeakers to the microphone, which sometimes can be heard as a characteristic ”howling” of tones with problematic frequencies for the specific enclosure.
Howling occurs when the microphone takes up too much of the loudspeaker energy and is undesired, resulting in an unpleasant experience for the audience and a risk of damaging the PA equipment.
The scenario can be described by the model shown in fig. 2.1. Throughout the work,
we will assume that the source signal u(t) contains speech only, the background noise
x(t)
G
y(t)
F
u(t)
Figure 2.1 – A model of the scenario case, here including one microphone and one loudspeaker (single-channel system).
will not be considered. Furthermore, the speech is assumed to have been sampled to the discrete domain at 16 kHz, which according to the Nyquist sampling theorem results in that all signals components up to 8 kHz will be sampled without aliasing[4]. The vast majority of the human speech is contained within this bandwidth, and therefore it is assumed that the continuous source signal is band limited to 8 Hz and thus can be sampled at 16 kHz and perfectly reconstructed from the samples without aliasing.
In fig. 2.1, a speaker produces speech into a microphone, resulting in a source signal u(t). The signal is then processed in the electro-acoustic forward path, here denoted G. This processing includes the amplifier gain and possibly digital audio effects such as compression and equalization. One of the most simple types of processing in the electro- acoustic forward path is a broadband gain, which is simply the ratio of the output signal power and the input signal power. A broadband gain G(t) can be expressed in dB as
Gain = 20log x(t) y(t)
!
[dB], (2.13)
and is the only processing in the electro-acoustic forward path considered in this work.
The amplified output signal x(t) is then transmitted to the loudspeaker. The output from the loudspeaker propagates through the room in which the PA system is set up, and interacts with the environment in a way described by the acoustic feedback path F . The acoustic feedback path is modelled as a linear system, with input signal x(t).
According to eq. (2.9), we can compute the output from that system, which is the
feedback signal going back into the microphone, by convolving the loudspeaker signal
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with the impulse response of the acoustic feedback path F (t), also denoted the Room Impulse Response (RIR). The signal is fed back into the microphone, forming a closed loop system described by
y(t) = F (t) ∗ x(t) + u(t)
x(t) = G(t) ∗ y(t), (2.14)
where F (t) and G(t) are the impulse responses of the acoustic feedback path and the electro-acoustic forward path, respectively. Upon computing the Fourier transform on both sides of eq. (2.14) and making use of the convolution theorem in eq. (2.11), one obtains:
Y (ω) = F (ω)X(ω) + U (ω) (2.15)
X(ω) = G(ω)Y (ω), (2.16)
where F (ω) and G(ω) are the frequency responses of the corresponding systems, and X(ω), U (ω) and Y (ω) are the frequency contents of their corresponding signal. From this, one can compute the total frequency response from the source u(t) to the output x(t) by using the property described in eq. (2.12):
H(ω) = X(ω)
U (ω) = G(ω)Y (ω)
Y (ω) − F (ω)X(ω) = G(ω)
1 − F (ω)G(ω) . (2.17) The term F (ω)G(ω) is referred to as the loop response of the system, and the related magnitude response |F (ω)G(ω)| is denoted the loop gain, whereas the phase response
∠F (ω)G(ω) is denoted the loop phase. The system described by the transfer function in eq. (2.17) is assumed to be a linear, time-dependent, finite order system, as described in section 2.1.1. These assumptions are justified in [3], where the authors argue that the linearity can be derived from the fact that a sound wave’s interaction with the environ- ment can be considered level independent, meaning that the nature of the reflections is not dependent on the sound pressure level and therefore linear. The time-dependency assumption is an obvious one, since the feedback path is dependent upon all movements and changes in the room, including the microphone or loudspeaker changing positions.
Finally, the system can be considered to be of finite order owing to the fact that RIRs in
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
−4
−2 0 2 4 6 8 ·10
−2Sample number
f (t )
Figure 2.2 – The impulse response of a typical room, truncated at 2001 samples
general are infinite, showing an exponential decay over time, as shown in fig. 2.2. From this observation, it is reasonable to allow truncation at a certain length of the RIR.
2.3 Stability analysis
Even though the system H(ω) is indeed time varying due to changes in the RIR, it is common practice in the field of feedback suppression to carry out the stability analysis for a time invariant system [3]. This is the reason that the expressions in eqs. (2.15) to (2.17) do not depend on time. The stability analysis originates from the paper ”Regeneration theory” by Harry Nyquist [5], which can be consulted for further reading. For the system described in eq. (2.17), the system becomes unstable for |F (ω)G(ω)| ≥ 1, or 0 dB. In order for the signal to diverge due to feedback, the components for the problematic frequencies from each loop needs to superimpose over time. For this to occur, the frequency components needs to be in phase, which requires the phase to be multiples of 2π. This condition for instability is summarized in the Nyquist stability criterion:
if there exists a radial frequency ω for which the loop gain is greater than or equal to
unity, and for which the loop phase is any multiple of 2π, then the system is unstable:
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0 1000 2000 3000 4000 5000 6000 7000 8000
−70
−60
−50
−40
−30
−20
−10 0
f [Hz]
|F(f)|[dB]
(a) The magnitude response
0 1000 2000 3000 4000 5000 6000 7000 8000
−700
−600
−500
−400
−300
−200
−100 0 100
f [kHz]
∠F(f)[rad]
(b) The phase response
Figure 2.3 – The characteristics of a typical room: the magnitude response and the phase response
|F (ω)G(ω)| ≥ 1 (2.18)
∠F (ω)G(ω) = m2π m ∈ Z. (2.19)
The corresponding frequency f = ω/2π will, if present in the source signal, cause unsta- ble oscillations in the system perceived as a howling sound. It should be pointed out that the assumption that the system is time-invariant is not necessarily fulfilled. Actually, it is virtually never fulfilled for any given PA scenario. However, under the assumption that the RIR is ”slowly changing” over time, the Nyquist stability criterion applies. It is important to note that this assumption can cause problems when the RIR is rapidly changing, such as when the speaker is holding a portable microphone and is walking around in the room, as explained in chapter 4. For this reason it is of importance to be aware to this assumption.
Any given room with RIR F (t) has a specific value of Maximum Stable Gain (MSG), which can be found from the frequency response F (ω). Expressed in dB, the initial MSG is computed by finding the peak with the largest magnitude in the frequency response that fulfils the phase condition eq. (2.19), and calculate how far that peak is from 0db.
The initial MSG is computed, in dB, as
− 20log max|F (ω)| ∀ω : ∠F (ω) = m2π m ∈ Z. (2.20)
The magnitude and phase responses of a typical room, and also one of the room char- acteristics used in the simulations in this work, are shown in fig. 2.3. The MSG of the RIR shown in fig. 2.3 is 3.087 dB.
The main objective of feedback suppression is to manipulate the total transfer function,
by introducing additional sub-systems which alters the total frequency response in order
to increase the MSG, preferably without distorting the source signal. In the following
chapters, we will look into different methods of achieving this.
3 Methods used in feedback suppression
This chapter is a summary of the history and available literature of the field of feedback suppression. The field of acoustic feedback suppression is a well studied subject, and several methods have been proposed to solve the howling problem. There are four main categories of feedback suppression, namely
• Periodic modulation methods
• Gain reduction methods
• Room modelling methods
• Spatial filtering methods (beamforming)
The first methods to address the issue of howling feedback, developed in the 1960s [1], [2], belong to the first category. Implemented with electronic components, these methods consists of manipulation of the microphone signal before amplification by altering the phase of the signal by a small value φ, or by shifting the frequency of the signal by a small
∆f . In [2], an increase in maximum stable gain of 14dB was reported, but the effects on the sound quality were too severe to be considered acceptable. Frequency shifting is a method that is used in some commercial products today. One of these methods, namely a frequency shift of 5 Hz, is evaluated in this work and is explained in-depth in section 4.1.
The second category, gain reduction methods, can be divided into three subcategories,
depending on the frequency range in which the gain is reduced. Early works applied a
full-band gain reduction upon detecting howling [6]. This method does obviously not
increase the maximum stable gain, but merely brings back an unstable system to a stable
state. Full-band gain reduction was later refined into Automatic Equalization (AEQ), which divides the input signal into frequency bands, and performs feedback detection on every sub-band. If a howling frequency is detected, the gain is reduced only in the sub-band where the critical frequency resides, thus leaving the rest of the signal intact.
The AEQ methods can be described as an attempt to automate the work of an audio engineer, who often works with sub-band equalization to reduce feedback. The AEQ method was further refined into Notch filter based Feedback Suppression (NFS), where notch filters are used to suppress problematic frequencies at which howling has been detected. Notch filters are stop band filters with a very narrow stop band (called a
”notch”), which severely reduces the gain in that particular frequency band and thus removes those frequencies from the signal. These notch filters can be designed to be very narrow, thus only suppressing a very small frequency band of the signal, namely where the howling occurs. It should be mentioned that notch filters can be implemented as both FIR and IIR filters, but in order to make them very narrow, a high order is required, which means that IIR filters are often prefered. To suppress several frequencies in a signal, a number of notch filters, centered at different frequencies, can be applied on a signal, either by applying several filters in series or by designing one filter with two or several ”notches”.
The NFS methods are by far the most used in commercial products today. All NFS
methods include a detection phase and a suppression phase [3], and are divided into
one-stage NFS methods and two-stage NFS methods. In one-stage methods, detection
and suppression are performed in the same step. In [7], the authors use adaptive notch
filters in order to detect and suppress howling in the same stage. It is concluded in
the paper, that the adaptive notch filters used in their work did not produce sufficient
feedback suppression in the entire frequency range. The most commonly used methods
in the NFS category are so-called two-stage methods, where detection and suppression
are separated. Often including the Fourier transform computed by the Fast Fourier
Transform (FFT), the frequency spectra of segments of the signal are evaluated. A
frequency spectrum is scanned with a peak-picking algorithm to find the frequencies
that has the most power, and the frequencies corresponding to these peaks are tested
against certain criteria to determine if they are indeed howling frequencies, or just tonal
components in the signal. If the detection algorithm finds a howling frequency, the
suppression stage receives information about the frequency at which howling occurs,
and applies a notch filter at that specific frequency to suppress the howling.
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There are several spectral (frequency based) and temporal (time-based) features that a howling components has, but a tonal component has not. In practice, one of these, or a combination of them can be used to determine if a peak in the frequency spectrum corresponds to a howling frequency. [8] evaluates a number of criteria that can be used to evaluate if a signal component is a howling component of a tonal component. A two-stage notch filter based method is evaluated in this work and is explained in-depth in section 4.2.
The third category, room modelling methods, sometimes also called Acoustic Feedback Cancellation (AFC), resembles the methods used in Acoustic Echo Cancellation (AEC), a feature used in conference telephony and other applications where a speaker commu- nicates with another speaker at a distant location using a conference telephone. In these cases, the far-end speaker’s voice is output from a loudspeaker and fed back into a mi- crophone, resulting in an echo back to the far-end speaker, if no measures are taken.
A common approach in AEC is to use an adaptive filter ˆ F to approximate the RIR F , and filter the output from the loudspeaker with ˆ F in order to model the feedback, and remove the approximated feedback component from the microphone signal. If the adap- tive filter is perfectly approximated, no feedback component remains in the microphone signal.
The main difference between the AFC case and the AEC case is that the loudspeaker signal is highly correlated to the microphone signal in AFC, which is not the case in the AEC case [9]. When there is high correlation between the loudspeaker and microphone signals, which occurs during ”double-talk” scenarios (when the near-end speaker and far- end speaker speaks simultaneously), the AEC algorithms are known to perform poorly in adapting the filters. This makes the AEC methods unsuitable for the AFC case, which can be described as the AEC case with constant double-talk. In order to use adaptive filters to remove the unwanted feedback, one needs to use decorrelation methods to decrease the correlation between the loudspeaker and microphone signals[3], [10], [11].
Different methods for decorrelation have been suggested, such as noise injection on the loudspeaker signal, frequency shifting or phase shifting the loudspeaker signal, non-linear processing, introduction of a delay in the forward path and decorrelating pre-filters[9].
The method of using adaptive filters to remove unwanted signal contents, and the AFC method evaluated in this work is further explained in section 4.3.
The fourth category, which is also known as beam forming, consists of using special
microphone- and/or loudspeaker arrays in order to reduce the signal transport between
the microphone and the loudspeaker, by modifying the directivity patterns of the array
to have the null direction in the direction of the other unit. These methods require additional hardware and will for that reason not be considered in this work, which is limited to software implementation.
In [3], the authors conclude that the most promising method in terms of achievable increase in MSG and subjective sound quality is the AFC approach. For this reason, one of these methods will be included in the MATLAB evaluation of methods. Upon surveying the market, it is obvious that the two-stage NFS methods are by far the most common in feedback suppression products. For this reason, one of these methods will be implemented and evaluated in MATLAB. The nature of these methods includes the disadvantage of being reactive, in the sense that howling sound needs to be detected, and thus is often heard before it is suppressed. This is clearly a drawback of these methods.
AFC on the other hand is a proactive suppression method, which removes feedback and
echoes continuously, making it slightly more interesting than the NFS approach. As
explained in the section above, the AFC methods need a routine for de-correlating the
loudspeaker signal from the microphone signal, and a 5 Hz frequency shift was chosen
for this, mainly due to its simplicity, but also since frequency shifting is by itself a
feedback suppression method, which then can also be included as a stand-alone method
for comparison. The algorithms, by which these three methods operate, are presented
in detail in the following chapter.
4 Description of algorithms
In this section, the three chosen methods frequency shifting, notch filter-based feedback suppression and acoustic feedback cancellation will be described in detail, and the nature of howling will be related to them.
4.1 Frequency shifting
The frequency shifting method, as the name suggests, manipulates the microphone signal by shifting all frequency components with a predetermined value ∆f . By performing this frequency shift, one aims at circumventing the magnitude condition eq. (2.18), by not allowing the signal components with the critical frequency f c to build up every loop, but instead being shifted to frequencies which fulfil the magnitude condition eq. (2.18), and thus stabilizing the system. A frequency shift can be performed in software by performing manipulations of the so-called discrete-time analytic signal
y a (t) = y(t) + iˆ y(t), (4.1)
where ˆ y(t) is the Hilbert transform of the original signal and i is the imaginary unit. The
analytic signal is defined as the original signal with zero negative frequency content. The
negative frequencies can be discarded due to the fact that audio signals are real signals,
and a property of real signals is that their frequency spectra is Hermitian, meaning that
the negative frequencies does not provide any information that cannot be found in the
positive frequency content [4]. One can perform frequency shifting by multiplying the
analytic signal with a complex exponential
x(t)
G
y(t)
F
FS u(t) d(t)
Figure 4.1 – The system with a frequency shift of the microphone signal in the electro- acoustic forward path
S mod (t) = e iω
st , (4.2)
where ω s = 2π∆f , and is the modulation frequency. The output from the frequency shift is then obtained by taking the real part of the resulting complex valued signal:
d(t) = Re(y a (t)S mod (t)) = y(t)cos(φ(t)) − ˆ y(t)sin(φ(t)), φ(t) = 2π∆f t, (4.3) where d(t) is the frequency shifted output signal. The modulation can be described by the system in fig. 4.1.
4.1.1 Analytic signal
The analytic signal can be obtained by computing the Fourier transform Y (ω) of a
segment of the input signal, and computing the inverse Fourier transform of the single-
sided spectrum, with the negative frequencies set to 0[12]. The inverse Fourier transform
is an approximation of the analytic signal. Since the spectrum of the approximated
analytic signal is single-sided, it is complex-valued and can be expressed according to
eq. (4.1). The nature of the Fourier transform requires that the input samples are
framed with frame size M samples which will introduce a delay of M samples in the
processing. In this work, an alternative method was used, which uses a modulated low
pass filter in order to obtain an approximation of the analytic signal [13]. To remove
the negative frequency components, a FIR low-pass filter of order 256 with normalized
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Figure 4.2 – The magnitude response of the modulated low-pass filter, with a pass band covering the entire positive frequency range and a stop band covering the negative frequency range.
cut-off frequency of f s /4 was used. This filter was modulated with the frequency f s /4, resulting in a complex-valued band pass filter with a pass band covering the entire positive frequency range, and a stop band covering the entire negative frequency range.
The filter is visualized in fig. 4.2.
The input samples were buffered into a delay vector of the same length as the complex modulated low pass filter (256), and for each sample, the dot product between the delay vector and the filter was computed in order to obtain the current analytic signal sample, which is an approximation of eq. (4.1) at time t. Equation (4.3) was then applied to the approximated analytical signal sample in order to obtain the frequency shifted output signal sample d(t).
4.2 Two-stage notch filtering
The two-stage notch filtering method makes use of information about the frequency
spectrum of the incoming signal in the detection stage, and applies notch filters to the
signal in the suppression stage, based on the findings in the detection stage. This section
describes the two-stage algorithm used in this work.
4.2.1 Detection stage
The incoming signal was framed in frames of M = 4096 samples using an overlap between frames to reduce detection time. The overlap was set to M/2 samples. When a frame had been filled with M samples, the frame was multiplied with a Blackman window to avoid spectral leakage. The frequency spectrum Y (ω) of the windowed signal was then computed with the Fourier transform. Due to the fact that the input signal is real-valued, it is sufficient to consider only the single-sided frequency spectrum, out of which the 10 largest peaks were located through a peak-picking algorithm. Evaluating 10 peaks gives a satisfactory level of confidence that a howling frequency is detected, since it is almost always the case that howling frequencies do not occur at the same level of applied gain, but occur ”once at a time” upon increasing the applied gain. In MATLAB, the function findpeaks was used for this. The frequencies corresponding to these peaks were considered ”possible howling frequencies” or {ˆ ω i }, 1 ≤ i ≤ 10.
The set of possible howling frequencies were then evaluated in three different steps to determine if the frequency at hand was a howling frequency or a tonal component of the input signal. This was done by the two spectral evaluations Peak-to-Harmonic Power Ratio (PHPR) and Peak-to-Neighouring Power Ratio (PNPR), along with the temporal evaluation Interframe Magnitude Slope Deviation (IMSD) [8].
4.2.1.1 Peak-To-Harmonic Power Ratio (PHPR)
Tonal components in speech often include harmonics, which are integer multiples of the frequency component. This is not the case for a howling frequency, which consists of a very narrow frequency without significant harmonics. The power of the possible howling frequency is divided by the power of the m ′ th harmonic to compute the PHPR. This feature is computed for each candidate howling frequency ˆ ω i for the m’th harmonic:
P HP R(ˆ ω i , m) = 10log 10
|Y (ˆ ω i )| 2
|Y (mˆ ω i )| 2 . (4.4)
4.2.1.2 Peak-To-Neighbouring Power Ratio (PNPR)
In speech, frequency components includes the property of having a broader bandwidth
than a single sinusoidal frequency component. In the frequency domain, this bandwidth
is identified by the power of the tonal component being shared over several neighbouring
frequency bins, centered around a peak. A howling component on the other hand, does
Acoustic feedback suppression
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not share power with the neighbouring frequency bins. By computing the power of a possible howling frequency and dividing it with the power of neighbouring frequency bins, one can make an assessment on whether the component is a tonal component or a howling component. The PNPR for the possible howling frequency ˆ ω i with the m’th neighbouring frequency bin is computed as
P N P R(ˆ ω i , m) = 10log 10
|Y (ˆ ω i )| 2
|Y (ˆ ω i + 2πm/M )| 2 . (4.5) The values computed in eqs. (4.4) to (4.5) are then compared to predetermined thresholds T P HP R , T P N P R , and if the computed values are higher than the threshold values for frequency ˆ ω i , it is considered to be a howling frequency.
4.2.1.3 Interframe Magnitude Slope Deviation (IMSD)
This feature uses the fact that howling has been observed to increases exponentially in energy over time, which means linearly in dB-scale. This increase is not observed in tonal components. IMSD for the possible howling component ˆ ω i computes a measurement of the deviation from linear increase, by performing differentiations between the energy for ˆ
ω i at older frames, and more contemporary frames. A large deviation from linearity, that is to say a large IMSD, suggests that the candidate is indeed not a howling frequency, whereas for small deviations, the candidate is considered a howling frequency. The IMSD is computed by
IM SD(ˆ ω i , t) = 1 M F
M
F−1
X
m=1
"
1 M F
M
F−1
X
j=0
1 M F − j
(20log|Y (ˆ ω i t − jP )| − 20log|Y (ˆ ω i , t − M F P |)−
1 m
m−1
X
j=0
1
m − j (20log|Y (ˆ ω i , t − jP )| − 20log|(Y /ˆ ω i , t − mP )|)
#
. (4.6)
The IMSD for each candidate howling component is compared to the threshold value
T IM SD , and if IM SD(ˆ ω i , t) < T IM SD , the frequency ˆ ω i is considered to be a howling
frequency.
Table 4.1 Threshold Value [dB]
T P HP R 10 T P N P R 30
T IM SD 1
4.2.1.4 Final assessment
The thresholds used in the three evaluations are presented in table 4.1.
For the PHPR, the 2nd and 3rd harmonics were included in the evaluation, and howling was said to be detected if the threshold was exceeded for all harmonics. In the PNPR, the six closest neighbours, three above and three below, were included, and howling was said to be detected if the ratio exceeded the threshold for all neighbours. The IMSD stored the frequency contents of the last 16 frames, and thus evaluated the slope for all possible howling frequency components over 16 frames. These numbers were inspired by [8], where the authors evaluated a number of spectral and temporal criteria for howling detection, and found the combination above to be robust and with a small false-alarm percentage 1 . The final threshold values were tweaked and tested until a reasonable howling detection was obtained.
The total assessment of the possible howling frequencies for each frame consisted of a combination of PHPR, PNPR and IMSD, and only if all three conditions for howling were fulfilled for the frequency ˆ ω i , it was considered to be a howling frequency, and actions were taken to suppress the frequency at hand.
4.2.2 Suppression stage
Upon detecting howling at frequency ˆ ω i , the suppression stage applied a notch filter in the acoustic forward path, centered at frequency ˆ ω i . A maximum of 20 notch filters was set, in order to prevent the source signal from being overly distorted. To make the filters as narrow as possible, biquadratic IIR filters were used in series, where the output y i [n]
from filter i with input x i [n] can be computed from the difference equation
1
