Sample rate less than Nyquist frequency?

[KevinHowJones]

In two gnuradio tutorials, one by Michael Ossmann[1] and one in the gnuradio wiki[2], examples are given where AFAICT, the sample rate used for the source block is less than 2 x the frequency being listened to, one actually quite a bit less. Both examples are listening to frequencies in the FM broadcast band (target 88.5 MHz and 97 MHz), but the sample rates used are 250 kHz and 10 MHz repectively. Yet the examples apparently work.

Please tell me what I am missing. It would seem to me that the sample rate would have to be 2 x 88.5 Msps or 2 x 97 Msps. I am quite new to DSP and SDR, so I am sure there is something I don't understand about all this.

[1]Software Defined Radio with HackRF, Lesson 1 - Great Scott Gadgets
[2]Guided Tutorial Hardware Considerations - GNU Radio

 

[DSheirer]

If the tuner were using direct sampling, yes, you'd be correct about the sample rate. Most tuners heterodyne the signal with a local oscillator-generated signal to mix the signal of interest down close to 0 MHz where it can be sampled at the lower sample rate.

For example, an FM radio station at 90.1 MHz can be mixed with a local oscillator at 89.0 MHz. This mixing would produce 2 copies of the original signal, one at (90.1 - 89.0 = 1.1MHz) and one at (90.1 + 89.0 = 179.1MHz). Run it through a low-pass filter to remove the higher frequency copy and you're left with the signal of interest at 1.1 MHz which can be sampled within the Nyquist rate.

 

[KevinHowJones]

If the tuner were using direct sampling, yes, you'd be correct about the sample rate. Most tuners heterodyne the signal with a local oscillator-generated signal to mix the signal of interest down close to 0 MHz where it can be sampled at the lower sample rate.

Ah yes, I see.

Is the beat frequency determined by the hardware, or the software? What I mean is, obviously the software has a say in this, but does the hardware just use a fixed difference based on the center frequency, or is the software explicitly telling it what to use, perhaps based on the sample rate?

I'm wondering this because if the hardware just always chooses a fixed frequency difference based on center frequency, one would have to be careful to choose a sample rate that would be at least double that. eg, if the IF is always going to be 1 MHz, you'd have to choose a sample rate of > 2 MHz.

Related, in Gqrx there is displayed a value called the "hardware frequency". I just calculated that to be 299.5 MHz from the center frequency (though that difference may vary.) Would that be the LO frequency?

 

[jonwienke]

Any SDR that isn't "direct sampling" heterodynes the input signal down to a lower frequency, and then digitally samples the lower frequency. The digitizer chip is always seeing the same input frequency, the only thing changing is the mixer frequency being added to the RF input. The software controls the mixer frequency being added to the input signal, but everything else tuning-related is determined by hardware.

 

[KevinHowJones]

Any SDR that isn't "direct sampling" heterodynes the input signal down to a lower frequency, and then digitally samples the lower frequency. The digitizer chip is always seeing the same input frequency, the only thing changing is the mixer frequency being added to the RF input. The software controls the mixer frequency being added to the input signal, but everything else tuning-related is determined by hardware.

Yes, I get that, but what I am wondering is how the value of the mixer frequency and thus output frequency is determined. Does the hardware just look at the center frequency from the software and target adjust the mixer frequency so it will always result in a given output frequency?

OR, does the software explicitly tell the hardware what mixer frequency to use, and if so, possibly base that on the sample rate chosen by the user?

The point is, I am wondering if one has to be careful about choosing the sample rate. IF the first scenario holds, and the hardware is always going to spit out a given output frequency, it would be good to know that is when writing the software so one could be sure to provide a high enough sample rate. But IF the software is explicitly telling the hardware what mixer frequency to use based on the chosen sample rate, one wouldn't have to be concerned with that.

 

[KevinHowJones]

On another point, I am wondering what happens to PSK or FSK modulated signals, say at UHF frequencies. If the signal is getting hetrodyned down to a very low frequency, does the information not get lost or distorted?

 

[KA1RBI]

if signals at the input of an ADC exceed the Nyquist frequency (1/2 sample rate), those signals don't simply vanish; instead they appear as "aliases". In such a case we may say that such signals are not in the first Nyquist zone. Provided of course that the input circuitry of the ADC doesn't filter them out, and provided that a zonal filter is placed ahead of the ADC to select the desired Nyquist zone (band), there is no reason why this can't be done.

I have a book (Multirate Signal Processing by fred harris) that describes this precise scenario: the implementation of an FM broadcast receiver (for the 88-108 MHz band) utilizing an ADC with a 80 MHz sampling rate...

73

Max

p.s. edited to correct sample rate = 80 MHz.

 

[ko6jw_2]

The Nyquist Theorem (actually the Nyquist-Shannon Theorem) is widely misunderstood and oversimplified. The idea that the sampling rate should be twice the highest frequency to be sampled needs to be qualified. It applies to a perfectly bandwidth limited signal which is not very likely in the real world. Thus, some sort of low pass filter is needed. Low pass filters never have a perfect cut-off. This results in aliasing as mentioned above. Also, Nyquist is the floor not the ceiling. Oversampling is commonly done perhaps at a rate several times the Nyquist frequency. This is done regularly with digital audio where the sampling rate could be as high as 384 Khz whereas CD's and MP3's use 44.1 Khz. It is an area of much debate with some saying that you can't hear the difference and others insisting that they can.

Nyquist's work was done in the late 1920's and concerned telegraph signaling. That is, the maximum signaling rate possible to resolve data at the receiving end. It actually had nothing to do with sampling. Shannon, in the late 1940's, applied Nyquist's theorem to sampling.

 

[KevinHowJones]

if signals at the input of an ADC exceed the Nyquist frequency (1/2 sample rate), those signals don't simply vanish; instead they appear as "aliases". In such a case we may say that such signals are not in the first Nyquist zone. Provided of course that the input circuitry of the ADC doesn't filter them out, and provided that a zonal filter is placed ahead of the ADC to select the desired Nyquist zone (band), there is no reason why this can't be done.

I have a book (Multirate Signal Processing by fred harris) that describes this precise scenario: the implementation of an FM broadcast receiver (for the 88-108 MHz band) utilizing an ADC with a 80 MHz sampling rate...

73

Max

p.s. edited to correct sample rate = 80 MHz.

Thanks for that. I'm not quite clear if you are answering my question regarding whether hetrodyning will cause FSK or PSK data to be lost, particularly if the output is being sent through a low-pass filter. Are you saying if they are aliased, they can be reconstructed or what?

Forgive if my naive take on this is just my novice understanding of SDR. But so far what I am understanding is:

In SDR, input signal is hetrodyned then low-pass filtered so lower sample rates can be used.
In this scheme, the audio from an FM broadcast can be extracted. I am assuming that for this to happen, the resultant frequency range from the signal passed from the low-pass filter would have to be that of the audio bandwidth, and the sample rate would have to be at least double that.

From what I gather from examples of SDR, other types of information is also extracted for hacking, some presumably ASK, PSK or FSK. If that is the case, is this also from data that was converted from a hetrodyned signal?

there is no reason why this can't be done.

There is no reason what can't be done?

Please forgive if I am not quite following things. The thing I am trying to understand with my question is, with the hetrodying -> low-pass filter scheme performed on a signal carrying PSK or FSK encoded data, will the data remain intact?

 

[jonwienke]

Please forgive if I am not quite following things. The thing I am trying to understand with my question is, with the hetrodying -> low-pass filter scheme performed on a signal carrying PSK or FSK encoded data, will the data remain intact?

Yes. That's how pretty much every radio ever made works, software-defined or conventional.

 

[KevinHowJones]

Yes. That's how pretty much every radio ever made works, software-defined or conventional.

Hmmm. Okay then, so carrying this logic further, could one take say a 2.4 GHz signal modulated with PSK (or QPSK), down-convert it (hetrodyne to a lower frequency) to say 900 MHz (which would be capable of propagating longer distance), then up-convert it back to 2.4 GHz on the receiver end, and have all the data remain intact?

 

[prcguy]

Yes, that works no problem. You would want to keep LO phase noise down in the conversion process and be aware of high or low side injection to avoid inverting the data, otherwise its an everyday thing.

BTW, all early first generation SDRs were a hetrodyne type converting the receive frequency to a low IF frequency due to the available A/D converters at the time (late 80s) and direct conversion for the VHF/UHF bands came later when hardware was more available.

Hmmm. Okay then, so carrying this logic further, could one take say a 2.4 GHz signal modulated with PSK (or QPSK), down-convert it (hetrodyne to a lower frequency) to say 900 MHz (which would be capable of propagating longer distance), then up-convert it back to 2.4 GHz on the receiver end, and have all the data remain intact?

 

[KA1RBI]

There is no reason what can't be done?

that provided the proper conditions are met, signals whose frequencies exceed one-half the sample rate can be correctly received, and that all the information contained therein can be accurately recovered.

In the FM receiver scheme that I laid out, there is *no* heterodyne. Instead, the signals are said to be "aliased to baseband".

with the hetrodying -> low-pass filter scheme performed on a signal carrying PSK or FSK encoded data, will the data remain intact?

Yes, with the caveat that the modulated signal that carries the PSK or FSK has a spectral bandwidth that must fit within the band defined by the low-pass filter.

take say a 2.4 GHz signal modulated with PSK (or QPSK), down-convert it (hetrodyne to a lower frequency) to say 900 MHz (which would be capable of propagating longer distance), then up-convert it back to 2.4 GHz

This is possible as long as the spectral bandwidth of the signal in the 2.4 GHz band will fit within the band limits imposed by the 900 MHz band, which are probably less wide than those on 2.4 GHz.

Normally however one would strongly consider demodulating/remodulating the PSK prior to repeating it (so as to regenerate the FEC - Forward Error Correcting codes).

73

Max

 

[KevinHowJones]

I have a book (Multirate Signal Processing by fred harris) that describes this precise scenario: the implementation of an FM broadcast receiver (for the 88-108 MHz band) utilizing an ADC with a 80 MHz sampling rate

Yikes! Must be a really good book...
($710 used from Amazon)

 

[KevinHowJones]

This is possible as long as the spectral bandwidth of the signal in the 2.4 GHz band will fit within the band limits imposed by the 900 MHz band, which are probably less wide than those on 2.4 GHz.

Yes, I figured there would have to be some limitation such as this.