# 0.2 The deconvolution method

 Page 1 / 1
The inverse filter approach to echo cancellation.

## Deconvolution

The output of a linear time-invariant (LTI) system is the convolution of the input signal with the impulse response of the system. If the classroom is modeled as an LTI system, then the output echoed signal, $y\left(t\right)$ , is the convolution of the input signal, $x\left(t\right)$ , with the room's impulse response, $h\left(t\right)$ .

The process of deconvolution involves designing an inverse filter ĥ(t) that is convolved with the echoed output signal to retrieve the original signal $x\left(t\right)$ . This can be done in either the time domain or frequency domain:

• Time domain: Use the deconv() method in Matlab on the echoed signal and the impulse response of the classroom in order to extract the de-echoed signal.
• Frequency domain: Take the Fast Fourier Transform (FFT) of both the impulse response of the room and the echoed signal. Point-wise divide the echoed signal by the transfer function, then take the inverse FFT of the result to extract the de-echoed signal.

However in practice, the system adds noise to input signal, meaning that if the signal-to-noise ratio is too low, the inverse filter will yield a noisy signal that poorly approximates the input.

## Finding the impulse response

A system is characterized at all frequencies by taking its impulse response . This is done by exciting a system with a Dirac delta function . The Dirac Delta Function is defined as:

$\delta \left(x\right)=\left\{\begin{array}{cc}+\infty ,& x=0\\ 0,& x\ne 0\end{array}\right\}$
$\underset{-\infty }{\overset{\infty }{\int }}\delta \left(x\right)dx=1$
The Dirac Delta Function is a distribution that is infinitely tall and infinitely narrow at 0, and the area under the Dirac Delta Function is defined to be 1.

However, it is practically impossible to excite a room with the ideal Dirac Delta function. Consequently, we used three methods to approximate the room's impulse response:

• Balloon Pop: filling a latex balloon with air and bursting it.
• Pseudo Dirac: using the dirac() function in Matlab to generate a vector of zeros with a single one in the center, then playing it using the sound() function.
• Sine-sweep Method:
1. Generate a logarithmically increasing sine signal over a desired frequency range (20 Hz to 20 KHz for this application)
2. Create an inverse chirp filter that time reverses the chirp and shifts it to become a causal signal (so that it exists in positive time). Then, divide the magnitude of the spectrum of the inverse filter by the square of the magnitude of the spectrum of the chirp signal.
The time shift inverts the phase of the chirp leading to linear phase after the convolution and the second set of operations neutralizes the squaring of the magnitude of the spectrum caused by the convolution.
• Convolve the chirp response of the room with the inverse chirp filter to get the impulse response of the room.

This logarithmically increasing sine signal can be characterized in the time domain by the following equation:

$x\left(t\right)=\mathrm{sin}\left[{\frac{T{\omega }_{1}}{\mathrm{ln}\left(\frac{{\omega }_{2}}{{\omega }_{1}}\right)}}^{\left({e}^{\frac{t}{T}\mathrm{ln}\left(\frac{{\omega }_{2}}{{\omega }_{1}}\right)}-1\right)}\right]$

where ${\omega }_{1}$ is the initial radian frequency and ${\omega }_{2}$ is the ﬁnal radian frequency of the sweep of duration $T$ .

## Sine sweep

To measure the response, we used:

• A small guitar amplifier with a relatively flat response facing the desks
• A directional cardioid microphone 2 meters from the amplifier at a thirty degree angle
• A microphone preamplifier
• An Apple Macbook Pro Laptop using Audacity for recording

## Problems

Several factors make this model unrealistic:

• Realistically, speakers move around and face different directions when they move. Our impulse responses are highly specific to the position of the signal's source. The information we get is based on the amplifier being stationary and facing a single direction.
• The impulse response is also time-varying. If even one person leaves, the average absorption coefficient of all the objects in the room changes and, per the Sabine equation, the reverberation time of the room also changes. People and objects are continually shifting, constantly affecting how the sound reflects around the room.

Time-varying adaptive filters are more sensible for problems such as suppressing reverberations in the OEDK classroom.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!