Loss of information and aliasing in superheterodyne receivers

Question

For a radio / SDR that downshifts a signal to an IF / intermediate frequency band, wouldn't information be lost in the case that the original signal is above the IF sampling rate? For example, the HackRF One has a 20Msps sampling rate, but a high-end frequency range of about 2GHz. If I sample a signal at 1.5GHz (for example) with the radio's sampling rate, shouldn't I lose information?

I know that I'm able to listen to FM broadcast radio on such a radio, though the signal I'm receiving is centered higher than the radio's sample rate - why is that?

Answer 1

Given a band limited signal, where there's no energy outside the frequency rate F_L to F_H, the Nyquist-Shannon sampling theorem tells us that there are two conditions you have to meet in order for your sampling process to be lossless:

1. The time period of a single sample has to be less than 1 / F_H.
2. The sampling rate must be at least 2 * F_H / N, and no more than 2 * F_L / (N - 1), where N is an integer, at least 1, and no more than F_H / (F_H - F_L).

Informally speaking, the first condition comes about because the time period of a single sample acts as a low-pass filter on the input signal. In the case of the HackRF One, the sampling part of the ADC takes a single sample in 1/6th of a nanosecond at most, giving you the 6 GHz upper limit. However, there's time taken after doing the sampling, before the ADC outputs the sample value.

So, given that the sampling process in the HackRF runs at at least 6 GHz, why doesn't it output samples that fast? Two likely reasons:

1. It can take a sample lasting 1/6th of a nanosecond, but then needs 49 nanoseconds to reset ready to sample again. This is unlikely at the HackRF One price point, but can happen.
2. The sampling phase can take samples that quickly, but the conversion from an analogue sample to a digital codeword takes longer. This is the normal reason for a fast sampler with a low sample rate.

The second condition is less intuitive; it's about what signals look like after sampling. Recall that we can consider a signal as a sum of sine waves via the Fourier transform, considering thus its frequency spectrum instead of its time value. A real-valued signal in the time domain has a spectrum that's mirrored around DC in the frequency domain - if it has a 1 kHz sine wave, then the frequency analysis will also show a -1 kHz sine wave of the same amplitude. We generally don't bother displaying the negative frequency waves - they are identical to the positive frequency waves in amplitude, but with the sign of the frequency negated.

When you sample such a signal, you get values that represent the waves. At this point, information is lost; the sampling theorem tells us that, for all sine waves of frequency "F + n * F_S" such that "0 ≤ F < F_S", where F_S is your sampling rate, and n is an integer in the rate -infinity to +infinity, you will get the same series of sample values. Remember that you can add up multiple sine waves to get a complicated signal, and their sample values simply sum up, so this works for more complex waveforms, as long as you can use the Fourier transform to express them as a sum of sine waves.

This is a problem for your receiver - with F_S being 20 MHz, but the band as a whole covering DC to 6 GHz after low-pass filtering by the sampler, a set of samples that "looks" like a 5 MHz wave might actually be anything from -5.995 GHz to 5.985 GHz in 20 MHz steps. And the negative frequencies have exactly the same signal content as their absolute value equivalents - so you have not just the 299 possibilities from 5 MHz to 5.985 MHz in 20 MHz steps, but also a set of possibilities that come from 15 MHz (abs(5 MHz - 1 * F_S) to 5.995 GHz in 20 MHz steps.

Fortunately, we can limit the set of possibilities with a band pass filter, which only lets through frequencies between a lower and an upper limit (and a set from the mirrored side of the filter). There's some complicated mathematics involved in considering the interaction of the sampling process and the filtering process[1], but as long as the lower limit of the filter is greater than the bandwidth of the filter, these interactions don't matter^[2].

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[1] And a neat trick involved, where you can look at the sampled output (discrete time) as-if it's been heterodyned down, because you get to choose the value of "n" when reconstructing the wave. So, if the filter only lets through frequencies from "5 * F_S to 5 * F_S + 15 MHz" (100 MHz to 115 MHz in your HackRF One), you can reconstruct with n = 1 instead of n = 5, and get the spectrum as-if you heterodyned down to F_S before sampling.

[2] The interaction between filtering and sampling involves some complicated maths, but briefly, an informal way to look at it is that after sampling, "DC" recurs every F_S, and the mirroring effect of the Fourier transform occurs at every one of these DC-equivalent points. If there's only one DC-equivalent point in the filter's pass band, then you can still determine which "side" of the mirror a given sample comes from, while if there's actual DC and a F_S multiple in the pass band, you have two mirrors in the pass band, and can't determine which mirror applied to a given signal.

Answer 2

Let us start with the receiver RX design path of the HackRF One.

The input signal is either mixed filtered with a 2170 MHz low pass filter(HPF for input > 2.7 GHz) or 2740 low pass filter(LPF for input < 2.14 Ghz) using the RFFC5072 with the IF frequency (2.4 - 2.6 GHz where the mixing products fall well outside the next stage due to the LPF of the next stage) or bypassed if close to the IF of the next stage.

//The IF up-conversion to 2.4 GHz is what lowers the price, mass produced WIFI ASIC chips, at WIFI 802.11g channel bandwidth(20MHz).

Then the signal passes to the MAX2837 or MAX2839(mostly similar) that takes the IF signal applies bandpass filtering and mixing that outputs a clean base-band, 1.75 to 28 MHz wide signal output set to D7:D4 = 1101 to match the 20MHz input of the next step.

That base-band filtered output is then fed to the ADC MAX5864 digitizing the resulting data stream to 8-bits at 22 MHz bandwidth. Fs = 44MHz.

When the desired signal falls within +- 10 MHz of the mixers tuned center frequency say, an FM Station of 200KHz bandwidth, is sampled above the Nyquist frequency, like 20 MHz, aliasing within +-10 MHz of the center frequency will not directly occur, depending on the channelization method and other signals present in the passband:

• Smoothed decimation, that produces reduced intensity +- images producing minor artefacts that are averaged to then spread across the bandwidth. (not used unless near real-time speed is a concern, faster but most inaccurate)

• Decimation then filtered("normal" method), decimation produces reduced intensity +- images which are then filtered.(most common in single receive applications.)

• Polyphase Channelizer then filters for each of the channels. Given the phase filtering bandwidth is much lower, smaller to start with, The Channelizer while not perfect, provides a much cleaner signal than above. Each channel must be filtered, but the magnitude (number of taps required) is much less due to the phase cancellation.

Sorry I digressed too far. I do not wish to de-fuzz too far.

When the Fs(44MHz from above) is greater than the Fs required, 400KHz, super sampling:

There are more points than needed to reconstruct the original signal. This is called over-sampling and can increase the range of signals beyond the 8-bit resolution of the A/D converter, by providing additional slopes of the signal to to be reconstructed with further processing, even if outside of the A/D's range normally clipped for over-voltage or the assumed voltage is 0).

The lower frequency information is still present in the signal until the tuner translates(via frequency shift) the signal to the appropriate range for the A/D. Remember the the IF range is +- bandwidth.

The complete answer to this question would require the mathematics behind the Shannon Limit. The simple take away is the more bits per second,the more bandwidth required. This has relation to frequency except higher frequencies provide more bandwidth.

There exists several Nyquist zones where signal recovery is possible for higher frequencies at a lower bit rate.

. For example in the case of a digital modulation wouldn't it be impossible to generate the same number of bits at a lower IF frequency, than at the original GHz-level freq?

No the number of bits possible for encoding is defined by bandwidth and not location.

The IF frequencies of your question are 2.3 - 2.7.

The same band-width as a 6 MHz NTSC video signal on 430 MHz,is the same size all communications < 6 MHz.

Yes, more bandwidth is available at higher frequencies.

If I sample a signal at 1.5GHz (for example) with the radio's sampling rate, shouldn't I lose information?

Yes you lose the information outside of sample range, +- 20 MHz.

Answer 3

For a radio / SDR that downshifts a signal to an IF / intermediate frequency band, wouldn't information be lost in the case that the original signal is above the IF sampling rate?

There's two different aspects of a signal at play here:

1. its center frequency
2. its bandwidth

We can start with this as an example:

I know that I'm able to listen to FM broadcast radio on such a radio, though the signal I'm receiving is centered higher than the radio's sample rate - why is that?

So for an FM broadcast you might have:

1. center frequency might be 89.100 MHz just for one example
2. bandwidth is usually only about 50 kHz (n.b. kilohertz)

This is very easy for the HackRF because it supports center frequencies anywhere from about 1 MHz to 6 GHz ✅, and it can fit many many such stations within its own bandwidth options from 2 Msps to 20 Msps ✅.

This is true for many other signals as well. Something like Bluetooth Low Energy (BLE) also is fine with the HackRF:

• center frequency approximately 2.4 GHz (✅ it's between 1 MHz to 6 GHz)
• bandwidth maybe something like 1 MHz (✅ well within 20 Msps maximum)

But not for something like 802.11ac WiFi because:

• center frequency approximately 5 GHz (✅ probably okay since < 6 GHz)
• bandwidth could be 80 MHz or more (❌ way beyond 20 Msps maximum)

If I sample a signal at 1.5GHz (for example) with the radio's sampling rate, shouldn't I lose information?

The frequency is within the range of the HackRF (✅) but you don't say the bandwidth (❓). So if you sample a ham radio SSB signal at that range (~3 kHz bandwidth ✅) you get all the information you need.

So maybe you tune the HackRF to 1499 MHz and its bandwidth to say 10 MHz, and then it samples everything from 1494–1504 MHz including your small SSB signal just fine. No information is lost e.g. the HackRF could just as easy reverse the process and send out all those original signals at their original frequencies.

But if you sample some really high bitrate or spread spectrum signal that covers 50 MHz of bandwidth you do lose information. Even though the center frequency is still fine (✅) the bandwidth is now too large (❌) and the HackRF "sees" (or "hears" might be better) less than half of the signal.