# Relation between Bandwidth, Bit Rate and Sampling Rate

Started by November 1, 2005
```Hi all,

Nyquist showed that minumum bandwidth, B required for baseband
transmission of Rs symbols per sec without ISI is B >= Rs/2

while for bandpass transmission of Rs symbols per sec without is
B >= Rs

So does it mean that if we use BPSK to transmit 500Mbps at baseband (or
passband)  then the we require at least 250MHz (or 500MHz) of
bandwidth?

Equivalently,  can we use 16-PSK to transmit 2Gbps (i.e. 4*500M symbol
per sec) at baseband (or passband) which requires us to use same
minumum bandwidth of 250MHz (or 500MHz) as in the case of BPSK?

If the above statements are right then it followed from sampling
theorem i.e.

For Baseband signal (low pass) -
The sampling rate must be greater than twice the highest frequency
compenent in the baseband signal.

For bandpass signal -
The sampling rate must be greater than twice the signal bandwidth,

Then, the sampling rate of the system also dependent on the symbol rate
of the system. Re-use the example above, the samplig rate for the
passband signal must be at least 2*500MHz=1GHz.

This sampling rate of 1GHz is sufficient for BPSK signal (where
symbol=bit) but how about the 16-PSK case. For 16-PSK system each bit
is now 0.5ns which is lower than the sampling time 1ns.
So can someone please let me know what are the problems?

Many thanks,
Regards,
Suterr

```
```suterr@gmail.com wrote:
> Hi all,
>
> Nyquist showed that minumum bandwidth, B required for baseband
> transmission of Rs symbols per sec without ISI is B >= Rs/2
>
> while for bandpass transmission of Rs symbols per sec without is
> B >= Rs
>
> So does it mean that if we use BPSK to transmit 500Mbps at baseband (or
> passband)  then the we require at least 250MHz (or 500MHz) of
> bandwidth?
>
> Equivalently,  can we use 16-PSK to transmit 2Gbps (i.e. 4*500M symbol
> per sec) at baseband (or passband) which requires us to use same
> minumum bandwidth of 250MHz (or 500MHz) as in the case of BPSK?
>
> If the above statements are right then it followed from sampling
> theorem i.e.
>
> For Baseband signal (low pass) -
> The sampling rate must be greater than twice the highest frequency
> compenent in the baseband signal.
>
> For bandpass signal -
> The sampling rate must be greater than twice the signal bandwidth,
>
> Then, the sampling rate of the system also dependent on the symbol rate
> of the system. Re-use the example above, the samplig rate for the
> passband signal must be at least 2*500MHz=1GHz.
>
> This sampling rate of 1GHz is sufficient for BPSK signal (where
> symbol=bit) but how about the 16-PSK case. For 16-PSK system each bit
> is now 0.5ns which is lower than the sampling time 1ns.
> So can someone please let me know what are the problems?
>
> Many thanks,
> Regards,
> Suterr
>

I'm not quite sure what your question is...
The bit time is relatively independent of the symbol time.  ie with
16-PSK you may have a bit time of 0.5nS into the encoder, but once
encoded and placed into the constellation the entire symbol now has a
2nS symbol time.  This is how the bandwidth is reduced for the same
encoded bit rate.  Thus the sampling only has to occur at 2x the
bandwidth (or 500MSPS), however it must have a high enough signal to
noise ratio to separate out the constellation points from each other
(and the noise).

In other words, by increasing the density of the encoded constellation
you may have reduced the bandwidth requirements, and therefore reduced
the sample rate requirement, but you have also increased the sampling
accuracy (signal to noise ratio) required.

This is all in sync with shannon's capacity ruling.  For a given
capacity if you increase the bandwidth you can decrease the required
signal to noise ratio, or for a narrower bandwidth you must increase the
signal to noise ratio.

As for the baseband/passband sampling, the rate must be equivalent, as
the same information is available in both situations.  Hence you only
need to sample the passband signal at twice the baseband bandwidth (just
as in the baseband case).  There is however a further requirement that
the centre passband frequency must be an integer multiple of the
sampling frequency.  Alternatively (as will most often be done due to
design constraints) the sampling frequency must be an integer divisor of
the centre passband frequency.

Hope that helps
```
```Hi Bevan,

> In other words, by increasing the density of the encoded constellation
> you may have reduced the bandwidth requirements, and therefore reduced
> the sample rate requirement, but you have also increased the sampling
> accuracy (signal to noise ratio) required.

Yes you are right and the equation related as follows
SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)

where k=log2(M) i.e. M is # of constellation. Rs and Fs are the symbol
rate and sampling rate, respectively.

So as the Fs decreases, the SNR will increase for a given Rb.
Similarly, if Rb decreases then the SNR will decrease as well. What
does it by decrease in SNR? It is the decrease in the required SNR
value to achiecve the same symbol error rate (SER) for the two
different rates? But I remember when I run a simulation for different
rates and plotted the SER vs. Eb/No, I obtained a strange result which
the higher data rate case is better than the lower data rate one.

Best Regards,
Suterr

```
```suterr@gmail.com wrote:
> Hi Bevan,
>
>> In other words, by increasing the density of the encoded constellation
>> you may have reduced the bandwidth requirements, and therefore reduced
>> the sample rate requirement, but you have also increased the sampling
>> accuracy (signal to noise ratio) required.
>
> Yes you are right and the equation related as follows
> SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)

umm... what?
Why would you want to put the equation like that?
And where did you get it from?

The sampling rate and symbol rate are quite separate from the signal to
noise ratio.  I mean there is a coupling (in terms of over-sampling
improving the signal to noise ratio a certain amount) but it's certainly
not as linear as this equation depicts.

The absolute basics are...  Sampling rate *must* be greater than twice
the symbol rate.  This means that it must be greater than the bit rate
Rb / log2(M) where M is the number of constellation points.  As the
number of constellation points (M) increases then the signal to noise
ratio must improve, as for a given transmitted bit energy there will be
less euclidean distance between each constellation point.  Hence any
noise present will cause a greater error rate, thus why TCM etc are very
popular in modern wireless communications.  They help to offset this
error rate by introducing error detection and correction in the
transmitted constellation, and thereby providing some 'encoding gain' to
offset the stricter requirements on SNR.

> where k=log2(M) i.e. M is # of constellation. Rs and Fs are the symbol
> rate and sampling rate, respectively.
>
> So as the Fs decreases, the SNR will increase for a given Rb.
> Similarly, if Rb decreases then the SNR will decrease as well. What
> does it by decrease in SNR? It is the decrease in the required SNR
> value to achiecve the same symbol error rate (SER) for the two
> different rates? But I remember when I run a simulation for different
> rates and plotted the SER vs. Eb/No, I obtained a strange result which
> the higher data rate case is better than the lower data rate one.

As Fs decreases, the SNR will be largely unchanged, until you begin to
alias, in which case everything will go to crap.
If Rb decreases, then the SNR will also stay the same, all that has
happened is that things are now slower, the constellation spacing is
still the same, because you haven't specified a different constellation
mapping, just a slower bit rate, and hence a slower symbol rate.
As the density of the constellation increases (i.e. we change the
constellation mapping) then the SNR of the system must increase to allow
us to distinguish between adjacent constellation points.

I have no idea why you plotted SER vs Eb/No... do you not understand the
correspondence between bits and symbols?
```
```Hi again,
> > Yes you are right and the equation related as follows
> > SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)

> umm... what?
> Why would you want to put the equation like that?
> And where did you get it from?
> The sampling rate and symbol rate are quite separate from the signal to
> noise ratio.  I mean there is a coupling (in terms of over-sampling
> improving the signal to noise ratio a certain amount) but it's certainly
> not as linear as this equation depicts.

The above equation is obtained from
http://www.mathworks.com/access/helpdesk/help/toolbox/comm/
then scroll dwon the left hand browser and search for
Channel
-->AWGN Channel
----->Decribing the Noise Level of an AWGN Channel
you will see the above equation.

Or if you go to the link below you will see some discussions also

A quick look on the digital communication text book also stated as
below
SNR=Eb/No * kRs/W where W is the bandwidth which is Fs must be at least
twice of W.

So is the above equation wrong?

> As Fs decreases, the SNR will be largely unchanged, until you begin to
> alias, in which case everything will go to crap.
> If Rb decreases, then the SNR will also stay the same, all that has
> happened is that things are now slower, the constellation spacing is
> still the same, because you haven't specified a different constellation
> mapping, just a slower bit rate, and hence a slower symbol rate.
> As the density of the constellation increases (i.e. we change the
> constellation mapping) then the SNR of the system must increase to allow
> us to distinguish between adjacent constellation points.
>
> I have no idea why you plotted SER vs Eb/No... do you not understand the
> correspondence between bits and symbols?

I am Sorry as I am a newbie in this communication area. So please bear
with me. I have been trying hard to understand this and I do

I think I understand the concept of bits and symbols as well as the
different modulation. However, I think I am very confused about getting
the correct noise variance for a particular Eb/No ratio for simulation.
Actually, I wanted to see the BER/SER performance of changing the data
rate (bit rate) say from 20Mbps to 100Mbps. But I guess I mess up
something here which I trying to track it down.

many thanks

Suterr

```
```suterr@gmail.com wrote:
> Hi again,
> > > Yes you are right and the equation related as follows
> > > SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)
>
> > umm... what?
> > Why would you want to put the equation like that?
> > And where did you get it from?
> > The sampling rate and symbol rate are quite separate from the signal to
> > noise ratio.  I mean there is a coupling (in terms of over-sampling
> > improving the signal to noise ratio a certain amount) but it's certainly
> > not as linear as this equation depicts.
>
> The above equation is obtained from
> http://www.mathworks.com/access/helpdesk/help/toolbox/comm/
> then scroll dwon the left hand browser and search for
> Channel
> -->AWGN Channel
> ----->Decribing the Noise Level of an AWGN Channel
> you will see the above equation.
>
> Or if you go to the link below you will see some discussions also
>
>
> A quick look on the digital communication text book also stated as
> below
> SNR=Eb/No * kRs/W where W is the bandwidth which is Fs must be at least
> twice of W.
>
> So is the above equation wrong?
>
> > As Fs decreases, the SNR will be largely unchanged, until you begin to
> > alias, in which case everything will go to crap.
> > If Rb decreases, then the SNR will also stay the same, all that has
> > happened is that things are now slower, the constellation spacing is
> > still the same, because you haven't specified a different constellation
> > mapping, just a slower bit rate, and hence a slower symbol rate.
> > As the density of the constellation increases (i.e. we change the
> > constellation mapping) then the SNR of the system must increase to allow
> > us to distinguish between adjacent constellation points.
> >
> > I have no idea why you plotted SER vs Eb/No... do you not understand the
> > correspondence between bits and symbols?
>
> I am Sorry as I am a newbie in this communication area. So please bear
> with me. I have been trying hard to understand this and I do
>
> I think I understand the concept of bits and symbols as well as the
> different modulation. However, I think I am very confused about getting
> the correct noise variance for a particular Eb/No ratio for simulation.
> Actually, I wanted to see the BER/SER performance of changing the data
> rate (bit rate) say from 20Mbps to 100Mbps. But I guess I mess up
> something here which I trying to track it down.
>
> many thanks
>
> Suterr

the BER/SER will not change if you increase the bit rate by increasing
the symobl rate IF the SNR stays the same.  However to pass a higher
symbol rate requires more bandwidth so to keep the SNR the same you
will need more power in the Tx.

If you increase the bit rate by keeping the symnbol rate the same but
instead increase the constellation points, then the BER/SER will degrde
beacuse the points are closer together so for a given noise power you
will have more errors.  Again you can increase the Tx power to maintain
the BER/SER.   Bottom line, raising the bit rate requires more Tx power
to obtain a given BER.  FEC coding  can reduce the BER at the expense
of some overhead, but in general (in nature) you don't get something
for nothing.

Mark

Look at the Shannon relationships.

Mark

```
```Hi Mark,

> the BER/SER will not change if you increase the bit rate by increasing
> the symobl rate IF the SNR stays the same.  However to pass a higher
> symbol rate requires more bandwidth so to keep the SNR the same you
> will need more power in the Tx.

I understand what you said. I mentioned above that I obtained results
from
two different symbol rate simulation and indeed it showed the same
thing you
explained here. But the SER vs EbNo is worse for the lower symbol rate,
would you be able to explain this abit to me?

Do you know the usage of the following equation
SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)
I think it is important for simulation setup but I don't want to use it
wrongly.

> If you increase the bit rate by keeping the symnbol rate the same but
> instead increase the constellation points, then the BER/SER will degrde
> beacuse the points are closer together so for a given noise power you
> will have more errors.  Again you can increase the Tx power to maintain
> the BER/SER.   Bottom line, raising the bit rate requires more Tx power
> to obtain a given BER.  FEC coding  can reduce the BER at the expense
> of some overhead, but in general (in nature) you don't get something
> for nothing.
Yes, I understand this one too....

Regards,
Suterr

```
```On 3 Nov 2005 07:41:58 -0800, suterr@gmail.com wrote:

>Do you know the usage of the following equation
>SNR(dB) = Eb/No(dB) + 10log10(k) + 10log10(Rs/Fs)
>I think it is important for simulation setup but I don't want to use it
>wrongly.

This equation makes some assumptions that may or may not be true.

Here's what is always true:

Eb = Ps - 10log(Rb)

No = Pn - 10log(BWn)

where Ps is the total signal power and Pn is the total noise power.
Rb is the _information_ (i.e., payload) bit rate and BWn is the
bandwidth of the noise.   Note that the symbol rate and the sampling
frequency don't enter into this anywhere.

SNR is simply SNR = Ps/Pn = Ps - 10log(Rb) - Pn + 10log(BWn)

So the equation you're using assumes no coding since it related Rs and
Eb directly via k in order to account for the modulation type.
That's fine if it's really the case.   It also assumes that the noise
bandwidth is equal to the Nyquist bandwidth which may or may not be
the case depending on whether you do any filtering in your simulation.

So there are some caveats with using the equation you've shown.  It'll
work fine if you manage those assumptions appropriately, but if the
noise isn't flat and isn't of the BW you expect then the results will
differ.   If there is coding overheard the results will differ.

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org
```
```Dear Eric,

> This equation makes some assumptions that may or may not be true.
>
> Here's what is always true:
>
> Eb = Ps - 10log(Rb)--------(1)
> No = Pn - 10log(BWn)--------(2)
OK

> where Ps is the total signal power and Pn is the total noise power.
> Rb is the _information_ (i.e., payload) bit rate and BWn is the
> bandwidth of the noise.   Note that the symbol rate and the sampling
> frequency don't enter into this anywhere.

but isn't Rb=k*Rs so that Rs is the symbol rate, k is the number of
bits per symbol
then can't we say that Eb=Ps-10log(Rb)=Ps-10log(kRs)?

> SNR is simply SNR = Ps/Pn = Ps - 10log(Rb) - Pn + 10log(BWn)
I don't really understand how we can arrive to the above equation. If
we substitute the early two equations i.e. (1) and (2) into the
SNR=Ps/Pn then we will obtain the same expression as SNR=Eb/No+10log(k)
-10log(Rs/BW) is this true?

> So the equation you're using assumes no coding since it related Rs and
> Eb directly via k in order to account for the modulation type.
> That's fine if it's really the case.   It also assumes that the noise
> bandwidth is equal to the Nyquist bandwidth which may or may not be
> the case depending on whether you do any filtering in your simulation.
> So there are some caveats with using the equation you've shown.  It'll
> work fine if you manage those assumptions appropriately, but if the
> noise isn't flat and isn't of the BW you expect then the results will
> differ.   If there is coding overheard the results will differ.
Yes I understand your points here...

Many thanks indeed.

You help me understand more and more.

Regards,
Suterr

```
```On 3 Nov 2005 23:58:15 -0800, suterr@gmail.com wrote:

>Dear Eric,
>
>> This equation makes some assumptions that may or may not be true.
>>
>> Here's what is always true:
>>
>> Eb = Ps - 10log(Rb)--------(1)
>> No = Pn - 10log(BWn)--------(2)
>OK
>
>> where Ps is the total signal power and Pn is the total noise power.
>> Rb is the _information_ (i.e., payload) bit rate and BWn is the
>> bandwidth of the noise.   Note that the symbol rate and the sampling
>> frequency don't enter into this anywhere.
>
>but isn't Rb=k*Rs so that Rs is the symbol rate, k is the number of
>bits per symbol
>then can't we say that Eb=Ps-10log(Rb)=Ps-10log(kRs)?

Yes, that's correct.

>> SNR is simply SNR = Ps/Pn = Ps - 10log(Rb) - Pn + 10log(BWn)
>I don't really understand how we can arrive to the above equation. If
>we substitute the early two equations i.e. (1) and (2) into the
>SNR=Ps/Pn then we will obtain the same expression as SNR=Eb/No+10log(k)
>-10log(Rs/BW) is this true?

Yes, that looks correct.   You can always check these relationships
for equivalence with a few examples.

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org
```