Mr Nyquist And His Aliases

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SDR expert Andrew Barron examines the relationship between the sampling rate used by an analogue-to-digital converter and the maximum bandwidth of signals that can be displayed on an SDR receiver’s panadapter.

 

SDR expert Andrew Barron examines the relationship between the sampling rate used by an analogue-to-digital converter and the maximum bandwidth of signals that can be displayed on an SDR receiver’s panadapter.

 

The Swedish-born American electronic engineer Harry Nyquist (1889-1976) determined the fundamental rules for analogue-to-digital conversion. He found that, as long as the sample rate was at least twice the rate of the highest frequency in the analogue signal, the original analogue signal could be recreated accurately from the digital data (Nyquist Theorem).

This means that, in order to accurately sample and then recreate, a band of radio signals from a few kHz up to 50MHz (fifty million Hertz) the analogue-to-digital-converter (ADC) should sample the radio spectrum at a minimum of 100Msps (one hundred million samples per second).

Higher ADC sampling rates mean that more bandwidth can be sampled, and a wider spectrum of frequencies can be displayed on the panadapter.

Most direct-sampling HF receivers use an ADC sample rate between 60 and 150Msps.

For example, a sample rate of 60 Msps allows the receiver to cover 0-30MHz.

And a sample rate of 150Msps means that the receiver can cover 0-75MHz.

By the way, an ADC reads one sample for every clock cycle, so the ADC sample rate in Msps is the same as the ADC clock frequency in MHz. Therefore, a direct-sampling HF radio has a frequency range equal to half the ADC clock rate and also half of the ADC sample rate.

 

Frequency Translation

The ‘dongle-type’ receivers and other SDRs that work above the HF bands, use a receiver chip or some other frequency translation stage, possibly an ordinary mixer and oscillator, to shift a band of frequencies down to a range that the ADC can sample.

In these cases, the ADC sample rate is often around 10Msps, so this type of SDR can display a band of frequencies that is 10 MHz wide. The frequencies displayed could be in the 2GHz range or even higher, but the panadapter is limited to the bandwidth set by the ADC sample rate.

Eagle-eyed readers will have spotted an anomaly here: The HF receiver could only display a maximum bandwidth of one half of the ADC sample rate, as set by the Nyquist theorem. But the receivers operating at higher frequencies can display a panadapter that is equal to the sample rate, and this seems to break the Nyquist rule.

The reason for this is that the data from the ADC is split into two 16-bit data streams known as the ‘I’ and ‘Q’ signals. The I and Q data is used to create the panadapter spectrum and the waterfall display.

It is also used to feed the DSP stage so that you can demodulate and hear the receiver or receivers that are displayed on the panadapter.

If you only had a single data stream, you would only be able to display 5MHz of the spectrum from a 10Msps ADC. However, by using both the I and the Q data streams, it is possible to accurately display 5MHz above the nominal centre frequency and 5 MHz below it.

This does not work for the HF receiver because, being direct conversion, the nominal centre frequency is zero. You can only display the frequencies above zero, up to the Nyquist frequency, which is half the ADC sample rate.

So far, we have discovered that having a higher ADC sample rate means that the receiver can display a wider bandwidth of the received spectrum. In a UHF or dongle-type SDR, the maximum-sized chunk of spectrum the receiver can display on a panadapter is equal the ADC clock or sample rate. In the direct sampling HF receiver, the chunk of spectrum that the receiver can cover is only half the ADC clock or sample rate.

 

Nyquist Zones

As explained above, a direct sampling HF SDR can receive signals from a few kHz above zero to one half of the ADC sample rate. This range of frequencies is called the First Nyquist Zone.  The range of frequencies from one half of the ADC sample rate up to the full sampling rate is called the Second Nyquist Zone.

There are more zones, increasing in frequency. For example, if the ADC is sampling at 100Msps, then the First Nyquist Zone is from 0-50MHz. The Second Nyquist Zone is from 50-100MHz and the third one is from 100-150 MHz. The ADC can sample frequencies well above the top of the first Nyquist Zone. For example, the ADC in an ANAN radio can sample frequencies up to around 700MHz.

So why don’t we build receivers that do that? The answer is that you can, but there are constraints. The main one being that you can only use one Nyquist Zone at a time.

 

Alias Frequencies

Take a piece of blank paper or a plastic sheet; something semi-transparent is ideal. Place it on a table, with the long side facing up (landscape format). Using a ruler and a pen, draw a line right across the page. Label the left edge of the line ‘0’ and label the right edge ‘200MHz.’

Now, fold the paper in half and, at the fold, label the line with ‘100MHz.’ Fold both sides back, so that you have a fan-fold (‘W’) pattern. Label the line on the new folds ’50MHz’ and ‘150MHz.’

On the first quarter, draw a computer screen that almost fits the range 0-50 MHz.

Fold the paper flat and draw an RF spectrum with interesting radio signals, like you have seen on images of panadapter displays, all the way across the paper.

You now have a model of an SDR system. Each quarter page is a Nyquist Zone. The signals in the 0-50MHz zone are displayed on the panadapter. Fold the paper, so that the Second Nyquist Zone overlays the first and hold the paper up to a bright light.

The RF signals in the Second Zone also appear on the panadapter, but they are reversed.

A signal at 80MHz appears at 20MHz on the panadapter. A signal at 60MHz appears at 40MHz. The actual radio signals are reversed as well, an upper sideband signal at 80MHz appears on the panadapter as a lower sideband signal at the 20MHz mark. This is not a problem for modes like AM and FM but is a concern for SSB.

If you overlay the Third Nyquist Zone, you will find that the signals are normal (not inverted).

The Fourth-Zone signals are inverted again.

With all four Zones overlaid, you will see a horrible mess of interference signals from 50-200MHz overlapping the wanted frequencies. The unwanted frequencies are called alias frequencies.

Obviously, it is very important that you stop the ADC from sampling the alias frequencies above the First Nyquist Band. Once a signal on an alias frequency has been sampled, it is impossible to remove it.

For this reason, all good HF SDR receivers will have a low pass filter before the ADC.

Here is the trick: If you replace the 50MHz low pass filter with a 50-100MHz bandpass filter, you can use the receiver to pick up signals in the 50-100MHz range. The panadapter will be backwards and the sidebands reversed, but this is very easy to fix in the SDR software. If you change to a 150-200MHz bandpass filter, the receiver can be used to receive 150-200MHz.

This effect is used on the Expert SunSDR pro receiver, enabling it to receive 80-160MHz. However, it is not commonly in use, probably because it is more effective to use a front-end frequency shifter or a receiver chip, like the dongle and other small SDR receivers do. That technique offers a smaller panadapter bandwidth over a much wider frequency range, which is generally more useful.

 

Front-End filters

Conventional super heterodyne HF (high-frequency) receivers have a bandpass filter before the first mixer stage. The filter is there to stop image signals appearing within the IF (intermediate frequency) passband, where they would interfere with the wanted band of signals.

Direct-sampling SDR receivers for the HF bands don’t have an ‘image problem’.

The direct conversion performed by the analogue-to-digital-converter means that direct-sampling SDRs don’t have a ‘hardware’ mixer and oscillator, so there are no image frequencies. But all SDR receivers should have a low pass or a bandpass filter in the front end, before the ADC (Fig. 1).

The standard configuration for a direct-sampling HF receiver is to have a low pass filter at around the Nyquist frequency placed just before the ADC. Signals in the First Nyquist Zone are sampled by the ADC and end up displayed on the panadapter.

Signals from above the First Nyquist Zone are not sampled because they are attenuated by the anti-alias low pass filter centred on the Fs/2 frequency (Fig. 2).

If you want to receive signals in the Second Nyquist Zone you can replace the low pass filter with a bandpass filter that covers the Second Nyquist Zone.

In the example drawing, the filter would pass frequencies from 50 to 100MHz. The 60MHz signal is reflected back, and it appears on the panadapter at the 40MHz position.

In a real SDR receiver, the panadapter would be adjusted (‘flipped’) to show 50 to 100MHz, and the inverted sideband would be corrected as well.

The Nyquist Zones are rather like a semi-transparent, fan-folded, paper map. If you look through page one, the second page folds over and is reversed. The third page, however, folds over the right way around (Fig. 3).

The drawing showing the Nyquist Zones illustrates the way that even-numbered Nyquist Zones are reflected or folded back in an inverted way, where the panadapter would appear to be reversed.

The odd-numbered zones are folded in the same way as Zone One; panadapter and sidebands are not inverted.

Note that, except for modes that only use a single sideband, or which carry different information on each of the sidebands, the sideband reversal does not matter. AM and FM are unaffected by the sideband reversal since the sidebands are identical.

SSB, digital modes carried as audio on a single sideband, ISB and some phase-shift modes would need to be corrected because USB becomes LSB. PSK31 will decode OK because the information is carried by the phase changes, but the operating frequency would be offset below rather than above the nominal ‘carrier’ frequency.

RTTY would be affected too, as it uses high and low tones, which would become reversed.

 

Filter Frequency and Response

The design of the front-end filter and the choice of its ‘roll-off’ frequency is quite important – especially when you are talking about an HF receiver.

The radio must be protected from the very high-power FM broadcast stations operating in the 88-104MHz range. Those signals are usually in the Second Nyquist Zone and – without a good anti-alias filter before the ADC – they can break through and be displayed on the panadapter display as wideband interference.

There is a temptation for SDR manufacturers to use as high a sample rate as they can, in order to get a receiver that can cover a wide frequency range.

If, for example, you use an ADC with a 160MHz clock, the Nyquist Zone ends at 80MHz, and this is perilously close to those large FM broadcast transmissions.

Unless the anti-alias filter has a sharp cut-off, there is a strong chance of getting interference from the FM band on the panadapter. If the anti-alias filter is at, or above, the Nyquist frequency, its roll-off can be plotted on the panadapter (Fig. 4).

The spectrum display image in Fig. 4 shows a mirror-image of the filter response of an SDR receiver with an 80Msps sample rate. To plot the shape of the filter on the panadapter display, you can use an RF signal generator to apply a fairly large signal to the receiver input. You sweep the frequency of the signal generator from the Nyquist frequency (half the ADC sample rate), up to a frequency that is 10-20MHz above that frequency.

The Second Nyquist Zone signals are reflected back onto the high-frequency end of the panadapter, revealing the filter response. This break-through of unwanted signals can be avoided if the anti-alias filter is tuned for a frequency that is lower than the Nyquist frequency. Yes, you lose some of the receiver’s possible frequency coverage, but any large signals above the Nyquist frequency are attenuated to a much greater degree, and they probably won’t appear on the panadapter.

The ANAN radios use this approach. Their ADC samples at 122.88Msps, giving a Nyquist frequency of 61.44MHz. However, the anti-alias filter is at 55MHz. So, signals at, and just above, the Nyquist frequency are well down the filter’s passband. If you run the test outlined above on an ANAN receiver, you do not see any unwanted signals reflected back onto the panadapter.

 

Next Month

This has been a rather technical article. I hope that it has shed some light on a couple of things that are different about SDR receivers, such as why it is good to have a high ADC sample rate and why the front-end filter is so important.

Next month, I will offer a review and my impressions of the Icom IC-7610 transceiver. It should arrive at my house in a few days, and I am excited to try it out.

 

[Andrew Barron ZL3DW is the author of ‘SDR: Software Defined Radio’ n(2nd ed., 2017) ISBN 9781 9101 9346 4); the title is available from the RadioUser website – Ed].

 

This article was featured in the November 2018 issue of Radio User