# Why different SFs in LoRa are said to be orthogonal?

As orthogonality is an important characteristic in LoRaWAN but most public references just make a conclusion that different SFs are orthogonal. This confused me a lot and hope to make a clear understanding.

As @Chris Stratton show me in the link http://www.sghoslya.com/p/lora_6.html, I can tell different signal from different SF and BW combination. But I still feel there's a gap that why this suggest orthogonal?

And the frequency-time figure is not the common frequency-amplitude figure that shows signal in frequency domain after Fourier Transformation. How this frequency-time figure used in real signal processing step? Also, in https://lab.dsst.io/slides/33c3/7945.html, the author suggested in page 80-83 the demodulation of LoRa. It leaves me the question that why up-chirp multiples the down-chirp result in a constant frequency?

I know from my previous study that the multiplication in time domain equals the convolution in frequency domain. So how does the de-chirp result make sense?

Thanks for @Neil_UK's kind suggestion: )

If you look at LoRa signals in the frequency-time plane on a waterfall spectral display, you will see a series of diagonal slashes which actually encode the data. The transmitter produces this, and the receiver has an algorithm which seeks to recognize something that looks like it, while ignoring anything that does not match the expectation.

LoRa is a propriety modulation scheme, and many of the details are not public, but when bandwidth is fixed, the fact that different spreading factors results in different data rates means it is reasonable to suppose that different spreading factors produces slashes of different slope (different rate of change of frequency vs. time). And diagonals with different slope will have only occasional intersection.

A little web searching turns up a page at http://www.sghoslya.com/p/lora_6.html which seems to endorse this view; however it additionally points out that the bandwidth setting is also a factor determining the slope. And from that makes the argument that there are combinations of spreading factor and bandwidth which would have the same slope as other combinations, and thus not be as fully orthogonal from them.

It's worth noting that when near a transmitter, the on-air behavior of LoRa signals can be easily displayed with a a low cost "RTL-SDR" dongle, though this would not work so well for weak signals from a remote transmitter. But up close, you could actually take a couple of transmitters and queue up packets at the same frequency and time, but different spreading factors, and see how they overlap but don't really interfere on the waterfal.

• Thanks for your kind and comprehensive reply. I read the page about LoRa orthogonality before and I see it's better to differentiate orthogonality between BW and SF combinations. Commented Jan 7, 2019 at 7:02
• (Feel sorry. I'm new to here and fresh to edit comment. This is the following up question of last comment. )I can see from the frequency-time figure suggests different signal. Here's my next question that why up-chirp multiples the down-chirp result in a constant frequency? I know from my previous study that the multiplication in time domain equals the convolution in frequency domain. So how does the de-chirp result make sense? Commented Jan 7, 2019 at 7:12

Different Spreading Factors (SFs) don't just happen to be orthogonal, they are designed to be orthogonal. The fact that they are orthogonal allows them to be easily separated in the receiver, hence the requirement for this feature.

There are many families of encoding that are orthogonal, sinewaves with integer numbers of cycles in a period, and Walsh functions, are just two examples. These both allow information to be spread over frequency and/or time.

• Thanks a lot. To be honest, I just follow the the orthogonality of "sinewaves with integer numbers of cycles in a period". As the time-amplitude figure can be translated to frequency-amplitude figure by Fourier Transformation, I have no idea what signal process step the frequency-time figure stands for? So I can not just see the differentiable signals from frequency-time figure and remember this is orthogonality. Hope for your kind guidance. Commented Jan 7, 2019 at 7:22
• @Armstrong I am not certain whether you've asked me a question, the phrase 'hope for your guidance' suggests you have. Is there something you've read that you're not sure about? If so, post a link to it in your orignal question (better than linking to it in a comment) and say what you do and don't understand about it, and I will try to be more specific. Commented Jan 7, 2019 at 8:55