LoRa integrate data into symbols by selecting the starting frequency of an up chirp. The resulting up chirp traverses across BW and arrives at the same place it started. At the demodulator, this chirp is multiplied by a down chirp followed by the fft to extract the data bits.
I tried to do this myself on paper but I can't help but find two different fft bins. I understand if both -ve & +ve bins are same, the demodulation is successful however, they don't aren't. I illustrate this at my best below.
As per DSP principles, the multiplication of received and locally generated base band chirps is the addition of the two instantaneous frequencies of the chirps in red and purple. Then, I have drawn a line which represents their summation in yellow. As we can see, the multiplication produces two products (in yellow), one positive and one negative frequency component that are not equal.
But according to literature , multiplication with a down chirp should produce only a single frequency bin, meaning, the positive and negative frequency products resulting from the multiplication must be of same magnitude. But, I see two different products here. How is that possible?
Would be so much helpful to me if someone can point out where I am wrong.