3.1 Lab 3: convolution and its applications  (Page 3/3)

Echo Cancellation

In this exercise, consider the problem of removing an echo from a recording of a speech signal. The LabVIEW MathScript function sound() or the function Play Waveform in LabVIEW can be used to play back the speech recording. To begin, load the .m file echo_1.wav provided on the book website by using the function wavread(‘filename’) . This speech file was recorded at the sampling rate of 8 kHz, which can be played back through the computer speakers by typing

>>sound(y)

You should be able to hear the sound with an echo. If the LabVIEW function Play Waveform (Functions → Programming → Graphics&Sound → Sound → Output → Play Waveform) is used to play the sound, you first need to build a waveform based on the loaded data and the time interval $\text{dt}=1/\text{8000}$ because this speech was recorded using an 8 kHz sampling rate. Connect the waveform to the function Play Waveform .

An echo is produced when the signal (speech, in this case) is reflected off a non-absorbing surface like a wall. What is heard is the original signal superimposed on the signal reflected off the wall (echo). Because the speech is partially absorbed by the wall, it decreases in amplitude. It is also delayed. The echoed signal can be modeled as $\text{ax}(t-\tau )$ where and $\tau$ denotes the echo delay. Thus, one can represent the speech signal plus the echoed signal as [7]

What is heard is $y(t)$ . In many applications, it is important to recover $x(t)$ – the original, echo-free signal – from $y(t)$ .

Method 1

In this method, remove the echo using deconvolution. Rewrite Equation (7) as follows [7]:

The echoed signal is the convolution of the original signal $x(\mathrm{n\Delta })$ and the signal $h(\mathrm{n\Delta })$ . Use the LabVIEW MathScript function deconv(y,h) to recover the original signal.

Method 2

An alternative way of removing the echo is to run the echoed signal through the following system:

Assume that $z[\mathrm{n\Delta }]=0$ for . Implement the above system for different values of $a$ and $N$ .

Display and play back the echoed signal and the echo-free signal using both of the above methods. Specify the parameters $a$ and $N$ as controls. Try to measure the proper values of $a$ and $N$ by the autocorrelation method described below.

The autocorrelation of a signal can be described by the convolution of a signal with its mirror. That is,

Use the autocorrelation of the output signal (echo-free signal) to estimate the delay time ( $N$ ) and the amplitude of the echo ( $a$ ). For different values of $N$ and $a$ , observe the autocorrelation output. To have an echo-free signal, the side lobes of the autocorrelation should be quite low, as shown in [link] .

[link] shows a typical front panel for this exercise. It is not necessary to obtain the same front panel but there should be controls for $a$ and $N$ as well as graphs to observe the echoed signal, echo-free signal and autocorrelation function of the echo-free signal.

Noise Reduction Using Mean Filtering

The idea of mean filtering is simply to replace each value in a signal with the mean (average) value of its neighbors. A mean filter is widely used for noise reduction.

Start by adding some random noise to a signal (use the file echo_1.wav or any other speech data file). Then, use mean filtering to reduce the introduced noise. More specifically, take the following steps:

1. Normalize the signal values in the range [0 1].
2. Add random noise to the signal by using the function randn . Set the noise level as a control.
3. Convolve the noise-added signal with a mean filter. This filter can be designed by taking an odd number of ones and dividing by the size. For example, a $1\times 3$ size mean filter is given by [1/3 1/3 1/3] and a $1\times 5$ size mean filter by [1/5 1/5 1/5 1/5 1/5]. Set the size of the mean filter as an odd number control (3, 5 or 7, for example).

Impulse Noise Reduction Using Median Filtering

A median filter is a non-linear filter that replaces a data value with the median of the values within a neighboring window. For example, the median value for this data stream [2 5 3 11 4] is 4. This type of filter is often used to remove impulse noise. Use the file echo_1.wav or any other speech data file and take the following steps:

1. Normalize the signal values in the range [0 1].
2. Randomly add impulse noise to the signal by using the LabVIEW MathScript function randperm . Set the noise density as a control.
3. Find the median values of neighboring data using the function median and replace the original value with the median value. Set the number of neighboring values as an odd number control (3, 5 or 7, for example).