# Why do SDR receivers emit I/Q samples?

To my understanding hardware receivers for software-defined radio applications basically take the input signal, mix it with the tuning frequency to remove the carrier frequency and then sample the resulting voltage with a sampling rate that is just high enough for the bandwidth of the payload signal. They emit those samples to the demodulation software in the form of I/Q value pairs. I'm assuming they obtain the Q value by taking another sample $1/4$ cycle (with respect to the tuning frequency) later, effectively doubling the sample rate.

Why do they use the I/Q representation?

I can see how I/Q is a nice representation (in hardware) when synthesizing signals because you can for example do frequency or phase modulation just by varying amplitudes, but this reason doesn't seem to apply to the case of SDR receivers.

So, is there anything gained by using I/Q for output instead of I at twice the sampling rate? Or is it just a matter of convention?

• @Gilles I rolled back your edit. It's really just one question phrased in multiple ways for clarity, styling them as an enumeration makes no sense to me. – AndreKR Feb 15 '17 at 11:06
• I answered a similar question over here: electronics.stackexchange.com/questions/39796/… – hotpaw2 Feb 15 '17 at 14:41

The SDR (or any general digital signal processing system) takes the received RF signal, and downconverts it from the carrier frequency to the baseband.

Now, the real bandpass signal from the antenna does not necessarily have a symmetric spectrum around the carrier frequency, but it can be arbitrary. If the downconverter now shifts the spectrum to the center frequency, the corresponding time-domain signal becomes complex. So, the I and Q samples you get from the SDR are the real and imaginary part of the complex baseband signal, that corresponds to your real passband signal around the carrier frequency.

More details can e.g. be found at the wikipedia site for digital down conversion.

I/Q representation does not correspond to different sampling points of the signal. Instead, it corresponds to the real and imaginary part of the digital complex-valued baseband signal. These parts are obtained by separately multiplying the RF signal with a sine and a cosine and sampling both streams after low-pass filtering.

Sampling with double frequency can yield the same information as I/Q. It would be neccessary to homodyne the signal to $f_s/4$ to get all information that would have been in the baseband IQ signal to be in the passband signal at $f_s/4$ (where $f_s$ is the sampling rate).

• Good answer. Just to clarify, I think you can yield the exact same information with sampling with a double frequency, if you allow the signal in IQ at baseband with sampling rate $F_s$ to exist at $F_s/2$ when sampling at $2F_s$ (in other words at 1/4 of the new sampling rate). Do you agree? – Dan Boschen Feb 15 '17 at 11:21
• @DanBoschen I believe you do not get the same information when you sample with double frequency from just a single stream (e.g. the multiplied with a sine). This will still just yield a real-valued baseband signal with double sampling rate, which corresponds to the even part of the baseband spectrum. Still, the odd part (i.e. the imaginary baseband signal) is not available. – Maximilian Matthé Feb 15 '17 at 11:49
• Consider that you can have the exact same spectrum at fs/4 that you can have at baseband (meaning the positive portion above fs/4 does not need to match the "negative" portion that in this case would be below fs/4). If you think about it, this is no different than having the real signal at the antenna (or carrier) representing the baseband IQ signal at DC. Although I haven't mathematically worked out the proof, but that is my thinking and recollection. – Dan Boschen Feb 15 '17 at 11:52
• Consider this example: A complex signal at baseband that is in the band less than +/- Fs/2, sampled at 2Fs. It is complex and it's positive spectrum from DC to Fs/2 is not the same as its negative spectrum from -Fs/2 to DC (and therefore requires two real signals whether it be I and Q or Magnitude and Phase to represent it). Now rotate that spectrum by multiplying by $e^{jnw \pi/2}$. where n is the sample count. The result will have shifted the spectrum to + Fs/4, with no spectrum in the negative half, but no other changes. Now take the real part. – Dan Boschen Feb 15 '17 at 12:00
• By taking the real part of the complex signal described above, a negative image will appear (Complex conjugate) and the the original signal will have scaled but is otherwise left unchanged. Apart from a scaling factor, the signal that was at fs/4 is identical to the baseband signal we started with; all the information is intact! (Just as when we move the signal to the carrier frequency which is also real). Do you see a flaw in my thinking? (I am also NOT implying using the "same" I as just twice the sampling rate, but meaning using just I which is a single real valued data stream). – Dan Boschen Feb 15 '17 at 12:04

There can be several reasons.

Computer processing:

One reason to use IQ data for SDR processing is to lower the computational processing rate (to use a slower or lower power processor) for visualization (panadapter) or demodulation without an additional conversion step. Many modulation schemes have asymmetric sidebands. IQ signals can carry disambiguated information about both sidebands around DC (0 Hz) (see explanation here), which means the processing rate can be very close to DC (0 Hz + signal bandwidth + filtering transition safety margin), as opposed to above twice the carrier frequency (plus signal bandwidth, filter transition band, and etc.). In fact, some SDR modules (Funcube Dongle Pro+, Elecraft KX3, etc.) produce IQ data into a PC stereo audio interface (thus allowing processing at very low audio data rates compared to much higher VHF/HF RF carrier or HF/LF IF frequencies).

To do processing with a single channel data stream requires either a very high processing rate (above 2X the RF carrier, using an FPGA, etc.), or some way to get rid of images or aliasing before downsampling/downconversion, usually by an additional conversion or mixing step (or more) to an IF frequency, plus one or more associated anti-aliasing filters for image rejection. Thus, a 2X rate single real data stream usually requires an additional IF stage (and/or a very narrow high frequency bandpass filter, often crystal or SAW) to do this compared to producing a 1X rate IQ data stream. An additional IF stage usually requires an additional oscillator and mixer as well. Whereas direct conversion to IQ data can be accomplished without the need for a high frequency band-pass or roofing filter for image rejection. And even double conversion to IQ data using an IF (intermediate frequency) often requires one less filter or one much lower quality filter (again, for equivalent image rejection).

The downconversion oscillator can be centered (or nearly so) on the signal carrier of interest (either RF or IF), or a low multiple, instead of being either offset or much higher. This can make tracking, phase locking, or synchronizing this oscillator simpler, and thus allow the frequency readout and/or transceiver transmitter signal generation to be simpler in minimal radio hardware.

Conversion hardware:

In hardware, it may be easier or cheaper to implement 2 ADCs at a lower sample rate, than 1 ADC at a higher sample rate. For instance, you can use a stereo sound card with a 44.1k (or 192k) sample rate, instead of a more expensive sound card with a 96k (or 384k) sample rate, for nearly the same signal bandwidth capability.

Chalkboard size:

IQ sample streams (created by two channels of 90 degree phase shifted mixing and/or sampling) also correspond closely to mathematical complex signals (with real and imaginary components), which makes it easier to think of the two channels of real data as one channel of a complex mathematical representation. This makes certain mathematical algorithms (DFT/FFT, complex envelope demodulation, etc.) more directly applicable (and, as mentioned above, at baseband processing rates) with less additional mathematical operations (offsets or fftshifts, etc.)

An explanation or description of these DSP algorithms using complex math usually requires less writing on a classroom chalkboard than equivalent explanations using a non-complex higher sample rate representation (as well as being far more elegant in the opinion of many.) These simpler complex/IQ explanations sometimes directly translate to less code (depending on the HLL computer language at its supported data types), or less computational blocks (using a graphical signal path design tool) is SDR applications.