Hi, I read on many resources that BER can be approximated as SER/(log2(M) at high SNR for awgn channel for M-QAM modulated transmission. Matlab simulations also verify this result. However, when I consider Rayleigh flat fading channel, BER is not equal to SER/(log2(M) . In fact, BER has a constant gap at high SNR. Can someone explain the recent and logic behind this ? I use Gray coding to modulate M-QAM symbols at both the channels. Thanks in advance. --------------------------------------- Posted through http://www.DSPRelated.com
Bit Error Rate vs Symbol Error Rate in Rayleigh fading M-QAM transmission
Started by ●June 9, 2015
Reply by ●June 9, 20152015-06-09
>Hi, > >I read on many resources that BER can be approximated as SER/(log2(M) at >high SNR for awgn channel for M-QAM modulated transmission. Matlab >simulations also verify this result. > >However, when I consider Rayleigh flat fading channel, BER is not equalto>SER/(log2(M) . In fact, BER has a constant gap at high SNR. > >Can someone explain the recent and logic behind this ? I use Gray coding >to modulate M-QAM symbols at both the channels. > >Thanks in advance. >--------------------------------------- >Posted through http://www.DSPRelated.comHi, I have an hypothesis (I'm not 100 % sure): First, where does the formula BER = SER / log2(M) comes from? It assumes that we are at medium to high SNR, and thus most of symbol errors are caused by **the confusion between two symbols which are neighbors** in the constellation (note: this is not true at low SNR). Since we suppose gray coding, two neighbor symbols differ only by one bit, so there is one bit error for each symbol error. So for n symbols, and m symbol errors, SER = m / n, and BER = m / (n * k), where k = log2(M) is the number of bits per symbol => BER = SER / log2(M). Now, if you are in a Rayleigh fading channel, then for periodic period of times you can expect big attenuation in the signal energy, e.g. very low SNR => the previous assumption (1 symbol error = nearest neighbour) is not true anymore. PS: Note that at the limit, at very low SNR, for each symbol error, you have about half the bit that are false too, and the formula becomes BER ~= SER / 2. With the Rayleigh channel, depending on your Doppler shift, you are between the two formula... I don't know if there exist a closed form relation depending on the max. Doppler shift and symbol rate. --------------------------------------- Posted through http://www.DSPRelated.com
Reply by ●June 9, 20152015-06-09
On Tuesday, June 9, 2015 at 12:04:44 AM UTC-5, Dan25 wrote:> Hi, > > I read on many resources that BER can be approximated as SER/(log2(M) at > high SNR for awgn channel for M-QAM modulated transmission. Matlab > simulations also verify this result. > > However, when I consider Rayleigh flat fading channel, BER is not equal to > SER/(log2(M) . In fact, BER has a constant gap at high SNR. > > Can someone explain the recent and logic behind this ? I use Gray coding > to modulate M-QAM symbols at both the channels. > > Thanks in advance. > --------------------------------------- > Posted through http://www.DSPRelated.comOn a Rayleigh flat-fading channel, by SNR is meant the **average** SNR (avSNR) where the averaging is over all possible fading levels. Since the (instantaneous)SNR is very very low when the channel is in a deep fade and the (instantaneous) BER is very nearly 0.5, the **average** BER (avBER) does not have the same relationship to avSNR as BER to SNR. For example, for BPSK on an AWGN channel, if we define SNR so that we can write BER = Q(\sqrt{2*SNR}) while on a Rayleigh fading channel, BER = 1/[2+avSNR] On an AWGN channel, BER decreases exponentially as a function of increasing SNR; on a flat-fading Rayleigh channel, avBER decreases only inversely a function of increasing avSNR, and a gap between the two functions is to be expected. Similar results hold for M-QAM systems with the additional (practical) complication that for M > 4, it is necessary to estimate the instantaneous SNR accurately so that the thresholds can be set correctly in the demodulators. Inaccurate settings of the thresholds has a strong effect on the instantaneous BER and hence on the actual avBER (as opposed to the theoretical avBER based on perfectly accurate thresholds). Nitpickers Anonymous: please note that SNR here is the BEND (bit energy to noise density) ratio. If you have other definitions of SNR, please make appropriate adjustments. Dilip Sarwate
Reply by ●June 9, 20152015-06-09
The BER formula for high SNR in AWGN is a correct approximation, for other channel conditions it is not a correct approximation. Exactly what more do you need to know? (If there's something specific someone here can probably help.) Steve
Reply by ●June 9, 20152015-06-09
dvsarwate <dvsarwate@yahoo.com> wrote:>Nitpickers Anonymous: please note that SNR here is the BEND (bit energy >to noise density) ratio. If you have other definitions of SNR, please >make appropriate adjustments. > >Dilip SarwateHi Dilip, Is "BEND" an official well-known acronym? Steve
Reply by ●June 10, 20152015-06-10
On Tuesday, June 9, 2015 at 12:12:47 PM UTC-5, Steve Pope wrote:> dvsarwate <dvsarwate@yahoo.com> wrote: > > >Nitpickers Anonymous: please note that SNR here is the BEND (bit energy > >to noise density) ratio. If you have other definitions of SNR, please > >make appropriate adjustments. > > > >Dilip Sarwate > > Hi Dilip, > > Is "BEND" an official well-known acronym? > > SteveBEND is an acronym that is very well-known to me, and I have used it several times in the past on comp.dsp. See, for example, a thread titled "DSSS-CDMA query" from about 5 years ago in which thread you also participated. I don't make a formal claim of priority, but I may well have invented the acronym myself! Whether it is official or not, I don't know and I don't care: I attribute my continued good health in retirement to my decision long ago to stay as far away as possible from Standards Committees, lexicographers, and Wikipedia authorship. Dilip Sarwate
Reply by ●June 10, 20152015-06-10
On Tue, 9 Jun 2015 20:14:56 -0700 (PDT), dvsarwate <dvsarwate@yahoo.com> wrote:>On Tuesday, June 9, 2015 at 12:12:47 PM UTC-5, Steve Pope wrote: >> dvsarwate <dvsarwate@yahoo.com> wrote: >>=20 >> >Nitpickers Anonymous: please note that SNR here is the BEND (bit energy >> >to noise density) ratio. If you have other definitions of SNR, please >> >make appropriate adjustments. >> > >> >Dilip Sarwate >>=20 >> Hi Dilip, >>=20 >> Is "BEND" an official well-known acronym? >>=20 >> Steve > >BEND is an acronym that is very well-known to me, and I have used it severa= >l times in the past on comp.dsp. See, for example, a thread titled "DSSS-CD= >MA query" from about 5 years ago in which thread you also participated. I d= >on't make a formal claim of priority, but I may well have invented the acro= >nym myself! Whether it is official or not, I don't know and I don't care: I= > attribute my continued good health in retirement to my decision long ago t= >o stay as far away as possible from Standards Committees, lexicographers, a= >nd Wikipedia authorship. > >Dilip SarwateBEND sounds like Eb/No. Is it the same thing? Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●June 10, 20152015-06-10
On Tuesday, June 9, 2015 at 11:49:08 PM UTC-5, Eric Jacobsen asked:> > BEND sounds like Eb/No. Is it the same thing? >Yes indeed, Eric, the BEND ratio is precisely Eb/N0. It is an important parameter that deserves a name all to itself. In my own mind, I think of it as the signal to noise ratio (SNR), but I don't name it so in public unless I state explicitly what I mean by SNR: as I did when I wrote "if we define SNR so that we can write BER = Q(\sqrt{2*SNR})" because the acronym SNR means whatever the reader thinks it means, and there are enough readers (and referees of my papers!) who insist, for example, that the BER for BPSK on an AWGN channel is given by Q(SNR) or (1/2)e^{-SNR} or whatever and anyone who says otherwise is in a state of sin. Besides, BEND ratio (or maybe just BEND) sounds much nicer than EBNO, don't you think? Dilip Sarwate President, AMBIT Corporation (AMBIT = Acronym may be ignored totally)
Reply by ●June 10, 20152015-06-10
On Wed, 10 Jun 2015 03:22:43 -0700 (PDT), dvsarwate <dvsarwate@yahoo.com> wrote:>On Tuesday, June 9, 2015 at 11:49:08 PM UTC-5, Eric Jacobsen asked: > >>=20 >> BEND sounds like Eb/No. Is it the same thing? >>=20 > >Yes indeed, Eric, the BEND ratio is precisely Eb/N0. It is an important par= >ameter that deserves a name all to itself. In my own mind, I think of it as= > the signal to noise ratio (SNR), but I don't name it so in public unless I= > state explicitly what I mean by SNR: as I did when I wrote > >"if we define SNR so that we can write BER =3D Q(\sqrt{2*SNR})" > >because the acronym SNR means whatever the reader thinks it means, and ther= >e are enough readers (and referees of my papers!) who insist, for example, = >that the BER for BPSK on an AWGN channel is given by Q(SNR) or (1/2)e^{-SNR= >} or whatever and anyone who says otherwise is in a state of sin. > >Besides, BEND ratio (or maybe just BEND) sounds much nicer than EBNO, don't= > you think?I'm torn. ;)>Dilip Sarwate >President, AMBIT Corporation > > > > >(AMBIT =3D Acronym may be ignored totally)Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●June 10, 20152015-06-10
On Wednesday, June 10, 2015 at 10:04:00 AM UTC-5, Eric Jacobsen wrote:> > I'm torn. ;) >I know, I know, me too. Especially since with SNR, we can create epigrams such as "BER for BPSK on an AWGN channel is given by Q(SNR) and anyone who says otherwise is in a state of sin. Love the sin; hate the SNR"
