# What happens if an harmonic in a band-limited signal at Nyquist frequency is added to a 90 degrees out-of-phase replica of it?

I'm interested about aliasing in digital audio and I wonder if aliasing can be produced by simple mixing of band-limited signals. As I know a band limited signal can contain frequencies up to sample rate/2 frequency, the so-called Nyquist frequency. Is this right?
If that is so, what will happen if an harmonic in a band-limited signal at Nyquist frequency is added to a 90 degrees out-of-phase replica of it? Will this create an harmonic with double frequency and could such an addition create aliasing?

• Possible duplicate of Can addition of two band limited signals create aliasing? – MBaz Mar 26 '18 at 15:42
• How is this question different from your previous one on the same topic? – MBaz Mar 26 '18 at 15:42
• @MBaz The other one does not address this question and I have already selected it as a correct answer. I tried to ask for this in a comment but got advised by the person who answered my other question to make a new one – user17127 Mar 26 '18 at 15:45
• BTW, recall that Nyquist requires signals to be below $f_s/2$. Harmonics at exacly the Nyquist frequency alias to DC. – MBaz Mar 26 '18 at 15:50

## 1 Answer

No. Adding two waves at the same frequency results in another wave of the same frequency. Depending on the phases, there can be anywhere from complete constructive interference to maximum destructive interference. If the amplitudes are the same there can be complete cancellation. This applies to the Nyquist frequency too.

See the math from my answer to FFT Frequency bin relationship.

Hope this helps.

Ced

• Thanks, if I add two sawtooth waves 90 degrees out of phase (in the time domain) don't I get a signal with double frequency? This confuses me – user17127 Mar 26 '18 at 15:54
• @John Am, In your question you asked about harmonics, which are pure tones. A saw tooth is a summation of harmonics, it already has the higher frequency tones embedded within it. – Cedron Dawg Mar 26 '18 at 15:57
• So in reality the summation just lead to different harmonics amplitudes and the fundamental does not change neither new harmonics are created. I think it is clear now. Thanks – user17127 Mar 26 '18 at 15:59
• @John Am, that is correct. It is even possible that the fundamentals cancel each other, and even the lower harmonics and all that is left is the higher harmonics. The frequency of the aggregate waveform will remain the same (unless something like all the odd harmonics are knocked out, but that would have to be a contrived case.) – Cedron Dawg Mar 26 '18 at 16:07