# Amplify a signal and phase shift it by multiplying by a complex number

Still need help 25th October 2019

I have a real time domain signal contained in an array, it's just two different frequencies summed together. If I want to amplify this signal by 10 and phase modulate it by 0.5 degrees, will this work?

1. Change the real signal to a complex signal. $$(e^{\omega_1t} + e^{\omega_2t})$$

2. Multiply it by the phasor $$Ae^{i \theta}$$

3. So the calculation is $$Ae^{i \theta} (e^{\omega_1t} + e^{\omega_2t})$$

4. Where $$A=10$$ is an amplitude change and $$\theta=0.5 \pi/180$$ is the phase change caused by the equipment

5. Take the real signal of this calculation to get the result

The numbers I have used, 10 and 0.5, are just an example.

• What is phase shift in time domain? Is not simply delay? – Creator Oct 13 '19 at 22:03
• en.wikipedia.org/wiki/Phase_modulation what does it say? You modulate message, generally message means a very specifc and time domain signal a broad meaning. – Creator Oct 13 '19 at 22:15
• @NatalieJohnson Please clarify, using equations if possible, what is the meaning of "phase modulate it by 0.5 degrees". That is not a term with a well-known definition in communications. – MBaz Oct 13 '19 at 23:38
• @NatalieJohnson both your approaches are wrong, but mainly due to the fact that you want to phase-shift a real-valued signal, but then expect it to still be real-valued – and it won't be! – Marcus Müller Oct 14 '19 at 7:40
• Ok, now we really need you to define what phase modulation means to you, in mathematical terms; having your signal model is great, by the way! All the info you're giving here in the comments should be worked into your question, so that a potential answerer doesn't have to read all our discussion here. – Marcus Müller Oct 15 '19 at 8:49

Multiplying by a complex number causes a rescaling by the magnitude and a rotation by the angle. This is why when you multiply two complex numbers, you multiply the magnitudes and add the angles.

The only signal for which a complex multiplication (rescale and rotation) is the equivalent of a phase shift is a single complex pure tone (looks like a spring, or threads on a bolt). The frequency is the translation value between the rotation amount and the shift size in the time domain. Thus if you have a signal that is a sum of complex pure tones with different frequencies, each will be shifted by a different amount by the same non-real multiplication factor, and the sum of the shifted tones will not match a shift of the sum.

Since a real pure tone is the sum of two complex pure tones of different frequencies, it cannot be shifted by a complex scalar multiplication, nor can any sum of them.

That's all there is to it.

• Hi, Yes I was testing on single frequency. Doesnt work for multiple frequencies. Do you know what would work? – Natalie Johnson Oct 21 '19 at 11:28
• @NatalieJohnson To add to Cedron's good answer- time delay is the negative derivative of phase with respect to frequency. So if you have a linearly decreasing phase as you increase frequency, the result will be a constant delay in time at all frequencies (hence the benefit of linear phase filters). – Dan Boschen Oct 26 '19 at 10:52