This example shows how to apply adaptive filters to signal separation using a structure called an adaptive line enhancer (ALE). In adaptive line enhancement, a measured signal x(n) contains two signals, an unknown signal of interest v(n), and a nearly-periodic noise signal eta(n).
The goal is to remove the noise signal from the measured signal to obtain the signal of interest.
Author(s): Scott C. Douglas
We'll first load in a signal of interest, a short clip from Handel's Hallelujah chorus.
audioReader = dsp.AudioFileReader('handel.ogg','SamplesPerFrame',44100); timeScope = timescope('SampleRate',audioReader.SampleRate,... 'YLimits',[-1,1],'TimeSpanSource','property','TimeSpan',1); while ~isDone(audioReader) x = audioReader() / 2; timeScope(x); end
You can listen to the signal of interest using the audio device writer.
release(audioReader); audioWriter = audioDeviceWriter; while ~isDone(audioReader) x = audioReader() / 2; audioWriter(x); end
Let's now make a periodic noise signal--a sinusoid with a frequency of 1000 Hz.
sine = dsp.SineWave('Amplitude',0.5,'Frequency',1000,... 'SampleRate',audioReader.SampleRate,... 'SamplesPerFrame',audioReader.SamplesPerFrame);
Now let's plot 10 msec of this sinusoid above. It shows 10 periods in 10 msec, just as it should.
eta = sine(); Fs = sine.SampleRate; plot(1/Fs:1/Fs:0.01,eta(1:floor(0.01*Fs))); xlabel('Time [sec]'); ylabel('Amplitude'); title('Noise Signal, eta(n)');
The periodic noise is a pure tone. The following code plays one second (one frame of 44100 samples) of the noise signal.
eta = sine(); release(audioWriter); audioWriter(eta);
The signal that we actually measure is the sum of these two signals, and we call this signal s(n). A plot of s(n) shows that the envelope of the music signal is largely obscured. Listening to a 3-second clip from the measured signal, the noise is clearly prominent...
release(audioReader); release(timeScope); release(audioWriter); count = 1; while count < 4 s = (audioReader() / 2) + sine(); timeScope(s); audioWriter(s); count = count + 1; end
An adaptive line enhancer (ALE) is based on the straightforward concept of linear prediction. A nearly-periodic signal can be perfectly predicted using linear combinations of its past samples, whereas a non-periodic signal cannot. So, a delayed version of the measured signal s(n-D) is used as the reference input signal x(n) to the adaptive filter, and the desired response signal d(n) is made equal to s(n). The parameters to choose in such a system are the signal delay D and the filter length L used in the adaptive linear estimate. The amount of delay depends on the amount of correlation in the signal of interest. Since we don't have this signal (if we did, we wouldn't need the ALE!), we shall just pick a value of D=100 and vary it later. Such a choice suggests that samples of the Hallelujah Chorus are uncorrelated if they are more than about 12 msec apart. Also, we'll choose a value of L=32 for the adaptive filter, although this too could be changed.
D = 100; delay = dsp.Delay(D);
Finally, we shall be using some block adaptive algorithms that require the lengths of the vectors for x(n) and d(n) to be integer multiples of the block length. We'll choose a block length of N=49 with which to begin.
The first algorithm we shall explore is the Block LMS algorithm. This algorithm is similar to the well-known least-mean-square (LMS) algorithm, except that it employs block coefficient updates instead of sample-by-sample coefficient updates. The Block LMS algorithm needs a filter length, a block length N, and a step size value mu. How do we pick mu? Let's start with a value of mu = 0.0001 and refine it shortly.
L = 32; N = 49; mu = 0.0001; blockLMSFilter = ... dsp.BlockLMSFilter('Length',L,'StepSize',mu,'BlockSize',N);
The output signal y(n) should largely contain the periodic sinusoid, whereas the error signal e(n) should contain the musical information, if we've done everything right. Since we have the original music signal v(n), we can plot e(n) vs. v(n) on the same plot shown above along with the residual signal e(n)-v(n). It looks like the system is converged after about 5 seconds of adaptation with this step size. The real proof, however, is obtained by listening;
release(audioReader); release(timeScope); release(audioWriter); while ~isDone(audioReader) x = audioReader() / 2; s = x + sine(); d = delay(s); [y,e] = blockLMSFilter(s,d); timeScope(e); audioWriter(e); end
Notice how the sinusoidal noise decays away slowly. This behavior is due to the adaptation of the filter coefficients toward their optimum values.
Now, removing a pure sinusoid from a sinusoid plus music signal is not particularly challenging if the frequency of the offending sinusoid is known. A simple two-pole, two-zero notch filter can perform this task. So, let's make the problem a bit harder by adding an FM-modulated sinusoidal signal as our noise source.
eta = 0.5 * sin(2*pi*1000/Fs*(0:396899)' + 10*sin(2*pi/Fs*(0:396899)')); signalSource = dsp.SignalSource(eta,... 'SamplesPerFrame',audioReader.SamplesPerFrame,... 'SignalEndAction','Cyclic repetition'); release(audioReader); release(timeScope); release(audioWriter); while ~isDone(audioReader) x = audioReader() / 2; s = x + signalSource(); timeScope(s); audioWriter(s); end
The "warble" in the signal is clearly audible. A fixed-coefficient notch filter won't remove the FM-modulated sinusoid. Let's see if the Block LMS-based ALE can. We'll increase the step size value to mu=0.005 to help the ALE track the variations in the noise signal.
mu = 0.005; release(blockLMSFilter); blockLMSFilter.StepSize = mu;
We now filter the noisy music signal with the adaptive filter and compare the error to the noiseless music signal.
release(audioReader); release(timeScope); release(audioWriter); while ~isDone(audioReader) x = audioReader() / 2; s = x + signalSource(); d = delay(s); [y,e] = blockLMSFilter(s,d); timeScope([x,e]); audioWriter(e); end
release(audioReader); release(timeScope); release(audioWriter);