aliasing - Generating a sound where end frequency is 1/2 of the start frequency

I am very very new to signals, and I am trying to generate sounds based on some parameters from a database. So, the database stores the following:

• startFrequency


• endFrequency


• startAmplitude


• endAmplitude


• duration

I am generating a WAV file based on this information using the following code:

...
double currentFrequency;
double currentAmplitude;

int samplerate = 44100;
int samples = ceil(duration * samplerate);
int* mySampleBuffer = new int[samples];

for(int i=0; i{

    currentFrequency = Interpolate(i, 0, samples, startFrequency, endFrequency);
    currentAmplitude = Interpolate(i, 0, samples, startAmplitude, endAmplitude);
    mySampleBuffer[i] = static_cast(INT_MAX * (sin(((double)i * currentFrequency * PI * 2)/(double)samplerate) * (currentAmplitude/maxAmplitude)));
}

writeWAVData("mySound.wav", mySampleBuffer, mySampleBufferSize, samplerate, 1);
delete[] mySampleBuffer;
...

When the currentFrequency reaches 1/2 of the startFrequency, I notice that the sound wave flips (not sure what the correct term for this is). It doesn't seem like this is correct. Could someone help me understand what is going on.

Here is a screenshot of Audacity

For this image, startFrequency is 3380 Hz, endFrequency = 1380 Hz, duration is 0.512 s, and the amplitude is constant from start to finish. I have calculated that at 0.43264 s, the currentFrequency is exactly 1/2 of the startFrequency. In the picture you can see the "flip" happen at about 0.43 seconds.

Answer

Think of frequency in this context more in terms of angular velocity. If that velocity is constant, nothing changes from the usual picture. However, if the frequency changes, it is more correct to compute the actual angle that is input into the trig functions as an anti-derivative of the frequency function.

Here the interpolated frequency is

$$f(t)=(1-\tfrac tT)\,f_0+\tfrac tT\,f_T,$$

so the angle at time $t$ is

$$\phi(t)=2\pi\,\int_0^t f(s)\,ds=2\pi\, [t(1-\tfrac{t}{2T})\,f_0+\tfrac{t^2}{2T}]=\pi t\,[(2-\tfrac tT)f_0+\tfrac tT\,f_T]$$

In contrast, your formula is

$$\phi_o(t)=2\pi\,t\,f(t)=2\pi t\,[(1-\tfrac tT)f_0+\tfrac tT\,f_T]$$

has a derivative or local angular velocity of

$$\phi_o'(t)=2\pi\,[(1-2\tfrac tT)f_0+2\tfrac tT f_T]$$

which has a zero at $t=\frac T2\,\frac{f_0}{f_0-f_T}$, which given your data is located at $t=0.43264s$.

Note that the correct formula also reaches the local angular frequency of zero, but at the double of that time interval, so outside $[0,T]$.

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Note on implementation: The last form of $\phi(t)$ can be constructed as

 pi*(i*dt)*(f0+interpolate(i,f0,fT,N))

with N=samples and dt=T/N, provided that

interpolate(0,f0,fT,N)==f0 and interpolate(N,f0,fT,N)==fT