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,
$y(t)$ , is the convolution of the input signal,
$x(t)$ , with the room's impulse response,
$h(t)$ .
The process of
deconvolution involves designing an inverse filter ĥ(t) that is convolved with the echoed output signal to retrieve the original signal
$x(t)$ . 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:
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:
Generate a logarithmically increasing sine signal over a desired frequency range (20 Hz to 20 KHz for this application)
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:
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.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
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
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?
Source:
OpenStax, Characterization and application of echo cancellation methods. OpenStax CNX. Dec 17, 2012 Download for free at http://cnx.org/content/col11468/1.3
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