Complex Fourier transforms and Quadrature detection

[Mark R]

Lets say I have a narrow band signal between about 100 MHz and 101 MHz and I want to perform a high-resolution spectral analysis.

Why could I not simply use a multiplier to generate an 'f - fr' signal - which, if fr = 100 MHz, would run from DC to 1 MHz?

Is it because the multiplier degrades the phase data of the original signal? If so, how could this be recovered by a quadrature detector, and used by the FT?

Do you simply get a 'real' and 'imaginary' output from the QD, which can passed directly as complex parameters to the FT?

 

[brentkiosk]

There must be someone more on top of this than I am, but you should be basically OK. Your system is essentially what's used in a TV or radio - an f-fr signal is produced over a frequency range convenient for processing. The final spectrum analysis is done by ear (for a radio) and the spectrum is not degraded by the mixing = multiplying process. This "spectral analysis" is only for amplitude, so I am not quite sure of phase info. But your mixing signal would have a well efined phase, frequency and amplitude, so should not cause degradation.

Hope someone else knows more.

 

[RaynorWolfcastle]

You're basically talking about a direct conversion (homodyne) receiver. I think, although I'm not positive, that 802.11b uses QPSK for channel coding, so they have a problem very similar to yours. You have a (relatively) narrowband IQ signal modulated at 2.4 GHz. The point is, homodyne receivers are used in practical systems, and it is gaining popularity for WCDMA handset front-ends (for a number of reasons which I won't get into).

There are a bunch of difficulties in making a good homodyne receivers. For one thing there tends to be leakage problems at the mixer, you get self-mixing and LO leakage effects that can cause DC offset issues. This system is also very sensitive to any frequency drift and phase noise in your LO as you can imagine. Phase error during the IQ demodulation can also be an issue. There are also noise-power-linearity issues that crop up in handsets.

Anyhow, your output from the receiver after demodulation is in fact two signals (or rather the IQ decomposition of one signal) which you can do whatever you want with. If you're really interested in any of this, I could look up my RF notes and give you a more detailed answer as to the challenges of direct conversion receivers.

P.S.: The more "conventional" way to downconvert a signal is through an intermediate frequency, although this also has it's share of drawbacks.

 

[djhuber82]

What do you mean by an 'f - fr' signal? You mean just multiply the original signal x(t) by cos(2*pi*fr*t)? If you did that, then becasue of aliasing you would end up converting both 100 - 101 MHz and 99 - 100 MHz to DC - 1 MHz, corrupting your signal if there happens to be another channel at 99 - 100 MHz. The Q required to filter out 99 - 100 MHz before mixing is unacheivable in practice. This is why direct conversion architectures use quadrature mixers; the quadrature outputs allow you to tell the difference between positive and negative frequency. In your example, a quadrature mixer could convert 100 - 101 MHx to DC - 1 MHz, while at the same time converting 99 - 100 MHz to -1MHz - DC.

 

[Mark R]

Perhaps my question was unclear.

I was trying to understand whether it is actually whether phase is preserved by a convetional balanced mixer. My interest is in the implementation of fourier zeugmatography, where preservation of the phase is required. My impression was that a mixer degrades the phase, hence the example below would not be possible using a simple mixer - and instead would require a quadrature mixer.

Take the scenario - I record a multi-frequency signal from time t - x, to t + x, with the property that at time t, all the components of the signal are at phase 0. (i.e. the signal begins with its components out of phase, they gradually drift into phase because of their differening frequencies, before drifting apart in the opposite direction).

I wish to repeat the recording multiple times, under varying conditions, so that some of the components begin to drift away from 0 phase at time t. However, I require that the subsequent demodulated signal accurately represent the phases of the individiual components between multiple recordings. The idea is to take these multiple recordings, 'stack' them up into a multidimensional array - and then perform multi-dimensional FT (1 dimension in the time domain, the other dimensions in the repetition domain). Clearly if the phase isn't maintained between experiments - then the FT in the higher dimensions will be meaningless.

 

[djhuber82]

Yes, mixing preserves phase. Why wouldn't it? Many wireless communications standards use phase-only modulation because a constant amplitude signal allows the use of a much more efficient power amplifier. If mixing didn't preserve phase then none of these radios would work.

 

[Mark R]

Doh! I can't believe I asked this question - the answer has all become clear.

Of course mixing preserves phase - but only provided that the phase of your LO remains constant. If you need to analyse data captured at multiple times (perhaps seconds or minutes apart), any drift in your LO will lead to phase drift in the mixer output.

As a qudarature detector directly returns the phase of the original signal, independent of the phase of the LO - the problem is solved.