I am trying to recognise a sequence of audio frames on an embedded system - an audio frame being a frequency or interpolation of two frequencies for a variable amount of time. I know the sounds I am trying to recognise (i.e. the start and end frequencies which are being linearly interpolated and the duration of each audio frame), but they are produced by a another embedded system so the microphone and speaker are cheap and somewhat inaccurate. The output is a square wave. Any suggestions how to go about doing this?
What I am trying to do now is to use FFT to get the magnitude of all frequencies, detect the peaks, look at the detection duration/2 ms ago and check if that somewhat matches an audio frame, and finally just checking if any sound I am looking for matched the sequence.
So far I used the FFT to process the microphone input - after applying a Hann window - and then assigning each frequency bin a coefficient that it's a peak based on how many standard deviations is away from the mean. This hasn't worked great since it thought there are peaks when it was silence in the room. Any ideas on how to more accurately detect the peaks? Also I think there are a lot of harmonics because of the square wave / interpolation? Can I do harmonic product spectrum if the peaks don't really line up at double the frequency?
Here I graphed noise (almost silent room) with somewhere in the interpolation of 2226 and 1624 Hz.
https://i.stack.imgur.com/R5Gs2.png
I sample at 91 microseconds -> 10989 Hz. Should I sample more often?
I added here samples of how the interpolation sounds when recorded on my laptop and on the embedded system.
https://easyupload.io/m/5l72b0
#define MIC_SAMPLE_RATE 10989 // Hz
#define AUDIO_SAMPLES_NUMBER 1024
MicroBitAudioProcessor::MicroBitAudioProcessor(DataSource& source) : audiostream(source)
{
arm_rfft_fast_init_f32(&fft_instance, AUDIO_SAMPLES_NUMBER);
buf = (float *)malloc(sizeof(float) * (AUDIO_SAMPLES_NUMBER * 2));
output = (float *)malloc(sizeof(float) * AUDIO_SAMPLES_NUMBER);
mag = (float *)malloc(sizeof(float) * AUDIO_SAMPLES_NUMBER / 2);
}
float henn(int i){
return 0.5 * (1 - arm_cos_f32(2 * 3.14159265 * i / AUDIO_SAMPLES_NUMBER));
}
int MicroBitAudioProcessor::pullRequest()
{
int s;
int result;
auto mic_samples = audiostream.pull();
if (!recording)
return DEVICE_OK;
int8_t *data = (int8_t *) &mic_samples[0];
int samples = mic_samples.length() / 2;
for (int i=0; i < samples; i++)
{
s = (int) *data;
result = s;
data++;
buf[(position++)] = (float)result;
if (position % AUDIO_SAMPLES_NUMBER == 0)
{
position = 0;
float maxValue = 0;
uint32_t index = 0;
// Apply a Henn window
for(int i=0; i< AUDIO_SAMPLES_NUMBER; i++)
buf[i] *= henn(i);
arm_rfft_fast_f32(&fft_instance, buf, output, 0);
arm_cmplx_mag_f32(output, mag, AUDIO_SAMPLES_NUMBER / 2);
}
}
return DEVICE_OK;
}
uint32_t frequencyToIndex(int freq) {
return (freq / ((uint32_t)MIC_SAMPLE_RATE / AUDIO_SAMPLES_NUMBER));
}
float MicroBitAudioProcessor::getFrequencyIntensity(int freq){
uint32_t index = frequencyToIndex(freq);
if (index <= 0 || index >= (AUDIO_SAMPLES_NUMBER / 2) - 1) return 0;
return mag[index];
}
imageData = new double*[imageHeight];
for(int i = 0; i < imageHeight; i++) {
imageData[i] = new double[imageWidth];
for(int j = 0; j < imageWidth; j++) {
// compute the distance and angle from the swirl center:
double pixelX = (double)i - swirlCenterX;
double pixelY = (double)j - swirlCenterY;
double pixelDistance = pow(pow(pixelX, 2) + pow(pixelY, 2), 0.5);
double pixelAngle = atan2(pixelX, pixelY);
// double swirlAmount = 1.0 - (pixelDistance/swirlRadius);
// if(swirlAmount > 0.0) {
// double twistAngle = swirlTwists * swirlAmount * PI * 2.0;
double twistAngle = swirlTwists * pixelDistance * PI * 2.0;
// adjust the pixel angle and compute the adjusted pixel co-ordinates:
pixelAngle += twistAngle;
pixelX = cos(pixelAngle) * pixelDistance;
pixelY = sin(pixelAngle) * pixelDistance;
// }
(this)->setPixel(i, j, tempMatrix[(int)(swirlCenterX + pixelX)][(int)(swirlCenterY + pixelY)]);
}
}
I am trying to implement a c++ function (code above) based on the following pseudo-code
which is supposed to create a swirl on an image, but I have some continuity problems on the borders.
The function I have for the moment is able to apply the swirl on a disk of a given size and to deform it almost as I whished but its influence doesn't decrease as we get close to the borders. I tried to multiply the angle of rotation by a 1 - (r/R) factor (with r the distance between the current pixel in the function and the center of the swirl, and R the radius of the swirl), but this doesn't give the effect I hoped for.
Moreover, I noticed that at some parts of the border, a thin white line appears (which means that the values of the pixels there is equal to 1) and I can't exactly explain why.
Maybe some of the problems I have are linked to the atan2 C++ standard function.
I watched this awesome video by Dave Miller on making a neural network from scratch in C++ here: https://vimeo.com/19569529
Here is the full source code referenced in the video: http://inkdrop.net/dave/docs/neural-net-tutorial.cpp
It uses mean squared error as the cost function. I'm interested in using a neural network for binary classification though and so would like to use cross-entropy as the cost function. I was hoping to add this to this code if possible, since I've already been playing around with it.
How would that be applied specifically here?
Would the only difference be in how the error is calculated for the output layer...or do the equations change all the way through backpropogation?
Does anything change at all? Is MSE versus cross-entropy solely used to get an idea of the overall error and not independently relevant to backpropogation?
Edit for clarity:
Here are the relevant functions.
//output layer - seems like error is just target value minus calculated value
void Neuron::calcOutputGradients(double targetVal)
{
double delta = targetVal - m_outputVal;
m_gradient = delta * Neuron::transferFunctionDerivative(m_outputVal);
}
double Neuron::sumDOW(const Layer &nextLayer) const
{
double sum = 0.0;
// Sum our contributions of the errors at the nodes we feed.
for (unsigned n = 0; n < nextLayer.size() - 1; ++n) {
sum += m_outputWeights[n].weight * nextLayer[n].m_gradient;
}
return sum;
}
void Neuron::calcHiddenGradients(const Layer &nextLayer)
{
double dow = sumDOW(nextLayer);
m_gradient = dow * Neuron::transferFunctionDerivative(m_outputVal);
}
void Neuron::updateInputWeights(Layer &prevLayer)
{
// The weights to be updated are in the Connection container in the neurons in the preceding layer
for (unsigned n = 0; n < prevLayer.size(); ++n) {
Neuron &neuron = prevLayer[n];
double oldDeltaWeight = neuron.m_outputWeights[m_myIndex].deltaWeight;
//calculate new weight for neuron with momentum
double newDeltaWeight = eta * neuron.getOutputVal() * m_gradient + alpha * oldDeltaWeight;
neuron.m_outputWeights[m_myIndex].deltaWeight = newDeltaWeight;
neuron.m_outputWeights[m_myIndex].weight += newDeltaWeight;
}
}
Finally found the answer here: https://visualstudiomagazine.com/articles/2014/04/01/neural-network-cross-entropy-error.aspx
You only have to change how the error at the output layer is calculated.
The relevant function to be changed is:
void Neuron::calcOutputGradients(double targetVal)
For mean square errors use:
double delta = targetVal - m_outputVal;
m_gradient = delta * Neuron::transferFunctionDerivative(m_outputVal);
For cross entropy just use:
m_gradient = targetVal - m_outputVal;
I'm trying to implement a very simple 1-dimensional gradient descent algorithm. The code I have does not work at all. Basically depending on my alpha value, the end parameters will either be wildly huge (like ~70 digits), or basically zero (~ 0.000). I feel like a gradient descent should not be nearly this sensitive in alpha (I'm generating small data in [0.0,1.0], but I think the gradient itself should account for the scale of the data, no?).
Here's the code:
#include <cstdio>
#include <cstdlib>
#include <ctime>
#include <vector>
using namespace std;
double a, b;
double theta0 = 0.0, theta1 = 0.0;
double myrand() {
return double(rand()) / RAND_MAX;
}
double f(double x) {
double y = a * x + b;
y *= 0.1 * (myrand() - 0.5); // +/- 5% noise
return y;
}
double h(double x) {
return theta1 * x + theta0;
}
int main() {
srand(time(NULL));
a = myrand();
b = myrand();
printf("set parameters: a = %lf, b = %lf\n", a, b);
int N = 100;
vector<double> xs(N);
vector<double> ys(N);
for (int i = 0; i < N; ++i) {
xs[i] = myrand();
ys[i] = f(xs[i]);
}
double sensitivity = 0.008;
double d0, d1;
for (int n = 0; n < 100; ++n) {
d0 = d1 = 0.0;
for (int i = 0; i < N; ++i) {
d0 += h(xs[i]) - ys[i];
d1 += (h(xs[i]) - ys[i]) * xs[i];
}
theta0 -= sensitivity * d0;
theta1 -= sensitivity * d1;
printf("theta0: %lf, theta1: %lf\n", theta0, theta1);
}
return 0;
}
Changing the value of alpha can produce the algorithm to diverge, so that may be one of the causes of what is happening. You can check by computing the error in each iteration and see if is increasing or decreasing.
In adition, it is recommended to set randomly the values of theta at the beginning in stead of assigning them to zero.
Apart from that, you should divide by N when you update the value of theta as follows:
theta0 -= sensitivity * d0/N;
theta1 -= sensitivity * d1/N;
I had a quick look at your implementation and it looks fine to me.
The code I have does not work at all.
I wouldn't say that. It seems to behave correctly for small enough values of sensitivity, which is a value that you just have to "guess", and that is how the gradient descent is supposed to work.
I feel like a gradient descent should not be nearly this sensitive in alpha
If you struggle to visualize that, remember that you are using gradient descent to find the minimum of the cost function of linear regression, which is a quadratic function. If you plot the cost function you will see why the learning rate is so sensitive in these cases: intuitively, if the parabola is narrow, the algorithm will converge more quickly, which is good, but then the learning rate is more "sensitive" and the algorithm can easily diverge if you are not careful.
The problem to solve is finding the floating status of a floating body, given its weight and the center of gravity.
The function i use calculates the displaced volume and center of bouyance of the body given sinkage, heel and trim.
Where sinkage is a length unit and heel/trim is an angle limited to a value from -90 to 90.
The floating status is found when displaced volum is equal to weight and the center of gravity is in a vertical line with center of bouancy.
I have this implemeted as a non-linear Newton-Raphson root finding problem with 3 variables (sinkage, trim, heel) and 3 equations.
This method works, but needs good initial guesses. So I am hoping to find either a better approach for this, or a good method to find the initial values.
Below is the code for the newton and jacobian algorithm used for the Newton-Raphson iteration. The function volume takes the parameters sinkage, heel and trim. And returns volume, and the coordinates for center of bouyancy.
I also included the maxabs and GSolve2 algorithms, I belive these are taken from Numerical Recipies.
void jacobian(float x[], float weight, float vcg, float tcg, float lcg, float jac[][3], float f0[]) {
float h = 0.0001f;
float temp;
float j_volume, j_vcb, j_lcb, j_tcb;
float f1[3];
volume(x[0], x[1], x[2], j_volume, j_lcb, j_vcb, j_tcb);
f0[0] = j_volume-weight;
f0[1] = j_tcb-tcg;
f0[2] = j_lcb-lcg;
for (int i=0;i<3;i++) {
temp = x[i];
x[i] = temp + h;
volume(x[0], x[1], x[2], j_volume, j_lcb, j_vcb, j_tcb);
f1[0] = j_volume-weight;
f1[1] = j_tcb-tcg;
f1[2] = j_lcb-lcg;
x[i] = temp;
jac[0][i] = (f1[0]-f0[0])/h;
jac[1][i] = (f1[1]-f0[1])/h;
jac[2][i] = (f1[2]-f0[2])/h;
}
}
void newton(float weight, float vcg, float tcg, float lcg, float &sinkage, float &heel, float &trim) {
float x[3] = {10,1,1};
float accuracy = 0.000001f;
int ntryes = 30;
int i = 0;
float jac[3][3];
float max;
float f0[3];
float gauss_f0[3];
while (i < ntryes) {
jacobian(x, weight, vcg, tcg, lcg, jac, f0);
if (sqrt((f0[0]*f0[0]+f0[1]*f0[1]+f0[2]*f0[2])/2) < accuracy) {
break;
}
gauss_f0[0] = -f0[0];
gauss_f0[1] = -f0[1];
gauss_f0[2] = -f0[2];
GSolve2(jac, 3, gauss_f0);
x[0] = x[0]+gauss_f0[0];
x[1] = x[1]+gauss_f0[1];
x[2] = x[2]+gauss_f0[2];
// absmax(x) - Return absolute max value from an array
max = absmax(x);
if (max < 1) max = 1;
if (sqrt((gauss_f0[0]*gauss_f0[0]+gauss_f0[1]*gauss_f0[1]+gauss_f0[2]*gauss_f0[2])) < accuracy*max) {
x[0]=x2[0];
x[1]=x2[1];
x[2]=x2[2];
break;
}
i++;
}
sinkage = x[0];
heel = x[1];
trim = x[2];
}
int GSolve2(float a[][3],int n,float b[]) {
float x,sum,max,temp;
int i,j,k,p,m,pos;
int nn = n-1;
for (k=0;k<=n-1;k++)
{
/* pivot*/
max=fabs(a[k][k]);
pos=k;
for (p=k;p<n;p++){
if (max < fabs(a[p][k])){
max=fabs(a[p][k]);
pos=p;
}
}
if (ABS(a[k][pos]) < EPS) {
writeLog("Matrix is singular");
break;
}
if (pos != k) {
for(m=k;m<n;m++){
temp=a[pos][m];
a[pos][m]=a[k][m];
a[k][m]=temp;
}
}
/* convert to upper triangular form */
if ( fabs(a[k][k])>=1.e-6)
{
for (i=k+1;i<n;i++)
{
x = a[i][k]/a[k][k];
for (j=k+1;j<n;j++) a[i][j] = a[i][j] -a[k][j]*x;
b[i] = b[i] - b[k]*x;
}
}
else
{
writeLog("zero pivot found in line:%d",k);
return 0;
}
}
/* back substitution */
b[nn] = b[nn] / a[nn][nn];
for (i=n-2;i>=0;i--)
{
sum = b[i];
for (j=i+1;j<n;j++)
sum = sum - a[i][j]*b[j];
b[i] = sum/a[i][i];
}
return 0;
}
float absmax(float x[]) {
int i = 1;
int n = sizeof(x);
float max = x[0];
while (i < n) {
if (max < x[i]) {
max = x[i];
}
i++;
}
return max;
}
Have you considered some stochastic search methods to find the initial value and then fine-tuning with Newton Raphson? One possibility is evolutionary computation, you can use the Inspyred package. For a physical problem similar in many ways to the one you describe, look at this example: http://inspyred.github.com/tutorial.html#lunar-explorer
What about using a damped version of Newton's method? You could quite easily modify your implementation to make it. Think about Newton's method as finding a direction
d_k = f(x_k) / f'(x_k)
and updating the variable
x_k+1 = x_k - L_k d_k
In the usual Newton's method, L_k is always 1, but this might create overshoots or undershoots. So, let your method chose L_k. Suppose that your method usually overshoots. A possible strategy consists in taking the largest L_k in the set {1,1/2,1/4,1/8,... L_min} such that the condition
|f(x_k+1)| <= (1-L_k/2) |f(x_k)|
is satisfied (or L_min if none of the values satisfies this criteria).
With the same criteria, another possible strategy is to start with L_0=1 and if the criteria is not met, try with L_0/2 until it works (or until L_0 = L_min). Then for L_1, start with min(1, 2L_0) and do the same. Then start with L_2=min(1, 2L_1) and so on.
By the way: are you sure that your problem has a unique solution? I guess that the answer to this question depends on the shape of your object. If you have a rugby ball, there's one angle that you cannot fix. So if your shape is close to such an object, I would not be surprised that the problem is difficult to solve for that angle.