# Digital Down Conversion

## 1 Down-Conversion

Down-conversion is a fancy name for pairing a mixer and a filter.

### 1.1 Mixing (a.k.a. Hetrodyning, Multiplying)

First, you mix (multiply) your input signal with a sine wave of a carefully selected frequency. As we know from the convolution theorem, multiplying in the time domain is the same as convolving in the frequency domain, so if our input signal is f(t) with Fourier transform $\hat f(\omega )$, we have

$\mathcal{F}\left[ f(t) e^{-i\omega _0 t}\right] = \hat f(\omega ) * \delta (\omega -\omega _0). \,\!$

By substituting $\omega ^\prime \equiv \omega -\omega _0$, so that

$\mathcal{F}\left[ f(t) e^{-i\omega _0 t}\right] = \hat f(\omega ^\prime +\omega _0) * \delta (\omega ^\prime ), \,\!$

it should become apparent that we’ve shifted the spectrum of f(ω). We now measure at frequency $\omega ^\prime$ the spectrum a $f(\omega ^\prime +\omega _0)$.

The only thing to pay attention to is that when I said “sine wave”, I really meant $e^{i\omega _0 t}=\cos (\omega _0 t) + i\sin (\omega _0 t)$. That is, you actually need to multiply f(t) by both a cosine (to get the real part) and a sine (to get the imaginary part) to have truly mixed with a tone at frequency ω0. This is commonly known as I/Q mixing (I = cosine, Q = sine). One of the hardest parts of I/Q mixing is ensuring that your cosine and sine waves are in perfect quadrature (i.e. phase shifted by exactly $\frac\pi 2$). Failure to do so will result in false “images” of tones at corresponding positive/negative frequencies.

If you are too cheap to mix with both components, and instead only mix with the cosine (or just the sine, it doesn’t matter), you’ve actually mixed with two tones: one at ω0 and one at − ω0. This we know from our basic trig identities for sine and cosine. As a result, $\omega ^\prime$ will now measure $\hat f$ at $\hat f(\omega ^\prime \pm \omega _0)$. If you are careful with your filters so that you get rid of the $\hat f(\omega ^\prime + \omega _0)$ part, then your cheapness didn’t cost you performance, and may have saved you some money.

### 1.2 Filtering

Once you mixed your band of interest so that it is near ω = 0, you can filter it to the bandwidth you are interested in. In general, it’s easier to make sharp filters at low frequencies. This is because filter performance scales with fractional bandwidth. Hence, getting a filter to roll of steeply within a 10-MHz range is easer if that range is at 10 MHz (where the fractional bandwidth is 1) than at 100 MHz (where the fractional bandwidth is 0.1).

Many concerns will drive your filter selection, including your science, digitizer sample rates, whether you need to suppress $\omega ^\prime + \omega _0$ components if you went with single-tone mixing, etc. One thing to be aware of is that, if you went with cosine/sine mixing, you’ll be needing two filters: one for the real component, and one for the imaginary. If these filters are not extremely well-matched, then you can quite easily change the relative amplitudes of the real and imaginary components. Since discriminating between positive and negative frequencies depends on the real and imaginary components having matched amplitudes, the effect of mismatched I/Q filters is to introduce false “images”, just like cosine/sine waves that aren’t in perfect quadrature.

## 2 Digital Down-Converters

Historically, radio-frequency (RF) signals that entered a telescope were down-converted in multiple steps. RF was first mixed (with a local oscillator (LO) and coarsely filtered to an intermediate frequency (IF) band before transmission back to a receiver, which would then mix and carefully filter again to down-convert the signal to baseband (a frequency band centered on zero).

However, with ADC sample rates reaching to higher and higher frequencies, and with the processing bandwidth in digital receivers increasing exponentially with Moore’s Law, it has become quite common to directly sample at IF, and to do the second down-conversion (if it is necessary at all) digitally. There are several benefits to doing this:

1. For I/Q mixing, you can guarantee quadrature because you generate cosine/sine waves digitally.
2. For I/Q filtering, you can guarantee that the filters are matched.
3. You can calculate exactly the filter response of your down-converter, and use that in your calibration.
4. After filtering (or while you are filtering, if you are clever), you can down-sample (reduce your sample rate) to stay at the critical Nyquist rate, and not waste resources processing unnecessary samples

As you can see, doing quadrature down-conversion loses almost all of its problems when it is done digitally, and that is a strong hint that most digital down-converters should be quadrature.