# How are signal bandwidth and MSPS related?

In various software defined radios, there are three important parameters to set when receiving: frequency, bandwidth and MSPS (million samples per second?). What does it mean to receive with a triple of parameters $(freq, b, s)$ (freq-frequency, b-bandwidth, s-MSPS)? Radio listens to frequencies in $[f-b/2,f+b/2]$ range, divides this range into $s$ chunks and transmits the sampled data to the PC?

How to calculate $s$ for a given $b$, to avoid aliasing?

Am I right that MSPS has nothing to do with the Nyquist theorem?

The need for modulation:

Your voice signals generally lies in the frequency range 1kHz to 4kHz and music lies in the range 20Hz to 20Khz. Say your locality has 4-5 AM broadcast stations. Now all the AM broadcast stations cannot transmit their content as is because if all the stations transmit at the same time there will be signal interference. Also, for efficient radiation of electromagnetic energy the radiating antenna should be of the order of the wavelength of signal radiated. If the signal is 20Hz-20kHz the length of antenna has to be of the size of kilometers. So, they place the information carried by 20Hz-20kHz at higher frequencies. Every station places its their content in their own band of frequencies. Like one AM station will operate on 100Mhz with their band of operation being 100Mhz - 20Hz and 100Mhz+20Khz and the second one at 105Mhz with their band of operation being 105Mhz - 20Hz and 105Mhz+20Khz and so and so forth. The process of placing the frequency content from one spectrum on to another is called modulations. Now this information is radiated in to free space by an antenna at the station.

The receiver has a tuning knob, which lets you select a band of frequencies. Say, you have tuned your receiver to receive station 3 which is operating at 110 Mhz. You will select all the frequencies in the range 110Mhz - 20Hz and 110Mhz+20Khz. The signal being present in the spectrum 110Mhz - 20Hz and 110Mhz+20Khz is brought back to 20 Hz to 20kHz using a process which reverses the operation in encoder side. This is called demodulation. This signal is ready to hear.

For a simple modulation-demodulation read this

Finally, if you want to post-process the signal before listening like reducing the noise, increasing bass/treble, you have to filter the signal.

In a software defined radio, there will be a RF receiver which does the job of the receiver mentioned above. The message signal that was recovered is then converted to digital samples using an analog-to-digital converter. The analog to digital converter has 2 steps, 1: Sampling at rate above or equal to Nyquist and 2. Quantization. These digital samples are processed using a software on a pc/embedded system. You can apply filters on these samples. These digital/discrete samples is then converted to analog using a D/A Converter. Main step of a D/A converter is interpolation.

Theoretically, in an ideal system, it is sufficient to sample the signal at Nyquist rate. Before play back, an ideal low pass filter will recover the message/baseband signal with out any information loss. But such a design is not feasible.

At a low sampling rate, the interpolation filter (low-pass filter) will not be able to interpolate intermediate samples effectively. However if the sampling rate is increased you can have better reconstructed message signal. Hence at higher msps you must receive a better quality signal than at low msps. However, the quality you perceive will become negligible after certain point as your ear might not be able to negotiate the difference.

I believe lowering the msps below Nyquist can introduce cross talk. That doesn't mean that you will receive neighbor radio stations because they got clipped at the RF receiver stage. The cross talk you receive is because of frequency warping.

In the above description i am speculating that sampling happens on the demodulated signal not on the modulated signal. Because demodulation can be easily achieved via an envelope detector for AM or a phase locked loop for FM. Demodulation in digital domain can be costly as sampling a high frequency signal at nyquist rate and operating at such high bandwidths can be resource consuming.

Also no FM station operates at Ghz. Giga is for satellite. 100 Mhz for FM. and Khz for AM.

• How is it possible to capture 1GHz signal with 3.2MSPS? That's the max. sample rate of RTL SDR. – user5631 Oct 10 '13 at 8:18
• The 1Ghz is the carrier signal. Around 1Ghz, in few Khz you will have the message signal. The RF antenna when tunes in to the station will demodulate the message signal from the carrier signal. I.e., the message is recovered from carrier. The process of A to D conversion happens on the message signal which is of the order Khz for voice/music stations. – Ram Oct 10 '13 at 10:40
• @user5631 First of all, the link you posted is not working. The radios capture modulated signals through the antenna and as ram said will be demodulated before sampling can be applied. sampling is always applied in the received side after the demodulation when there is only message signal left. The USRP N210 series has a sampling rate of 100MSPS.So theoretically that means it can sample about 50 MHz of bandwidth. But then your computer processor has to be really fast to sample at that rate which is unlikely.So practically we achieve about 10 Mhz bandwidth signals using SDR, – Karan Talasila Oct 10 '13 at 13:51
• No system can sample a 1GHZ signal. It is impossible to design ADC's to sample at such high rate. When they talk about radios capturing modulated signals, it means the daughter board and antenna will receive a modulated electromagnetic signal at those frequency ranges. The sample rate is applied only after you get the message signal. You have to first understand the difference between bandwidth and carrier frequency. That will help you understand your confusion. – Karan Talasila Oct 10 '13 at 13:56
• I checked the link. It is clearly given that radio frequency range for example USRP B 210 is 50Mhz-6GHz which means that the radio can operate on modulated signals in range of 50Mhz-6GHz and not message signals of bandwidth 50Mhz-6GHZ, That's the difference you need to understand. Try doing one thing. on the USRP B210 try to send a signal in maybe 3khz. The flowgraph will not work and it will say error because transmission is not in frequency range. That is because you have to modulate the signal into the frequency band of operation and send it for the radio to work. – Karan Talasila Oct 10 '13 at 14:30