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# Bit Error Rate (BER) for frequency shift keying with coherent demodulation

by on August 30, 2007

Following the request by Siti Naimah, this post discuss the bit error probability for coherent demodulation of binary Frequency Shift Keying (BFSK) along with a small Matlab code snippet.

Using the definition provided in Sec 4.4.4 of [DIG-COMM-SKLAR]), in binary Frequency shift keying (BFSK), the bits 0′s and 1′s are represented by signals $s_1$ and$s_2$ having frequencies $f_1$and$f_2$ respectively, i.e.

$s_i(t) = \sqrt{\frac{2E}{T}}cos(2\pi f_it + \phi)$,

where

$E$ is the energy ,

$T$ is the symbol duration and

$\phi$is an arbitrary phase (assume to be zero).

The two frequencies $f_1$and$f_2$ are orthogonal, i.e.

$\int_0^T\sqrt{\frac{2E}{T}}cos(2\pi f_1t + \phi)\sqrt{\frac{2}{T}}cos(2\pi f_1t + \phi)dt=\sqrt{E}$ and

$\int_0^T\sqrt{\frac{2E}{T}}cos(2\pi f_1t + \phi)\sqrt{\frac{2}{T}}cos(2\pi f_2t + \phi)dt=0$.

Simple transmit-receive block diagram for binary frequency shift keying (FSK) can be as shown below.

Figure: Block diagram of FSK modulation and coherent demodulation

For analyzing the bit error rate with coherent FSK demodulation, let us compare the signaling waveform used by binary FSK when compared with binary PSK. The distance between the energy of the signaling waveform for:

(a) binary phase shift keying (BPSK) is $4E_b$ (uses antipodal signaling)

(b) binary frequency shift keying (BFSK) is $2E_b$ (uses orthogonal signaling)

Figure: Orthogonal and antipodal signaling

Using similar mathematical formulation used for BPSK, but with the distance between the signals reduced by half, the bit error probability for coherent binary frequency shift keying is

$P_b = \frac{1}{2}erfc\left(\sqrt{\frac{E_b}{2N_0}}\right)$.

For obtaining the same bit error rate as BPSK, binary frequency shift keying requires around 3dB more $\frac{E_b}{N_0}$. More details on the bit error curves with orthogonal and antipodal signals is discussed in Sec 3.2.5 of [DIG-COMM-SKLAR].

## Simulation Model

Simple Matlab/Octave script for computing the bit error rate with FSK modulation. The code performs the following:

(a) Generation of random 1′s and 0′s

(b) Converting bits to appropriate frequency

(c) Passing through Additive White Gaussian Noise channel

(d) Demodulation at the receiver

(e) Counting the number of errors.

Click here to download Matlab/Octave script for computing Bit Error Rate with FSK modulation

Figure: Bit error probability with coherent demodulation of frequency shift keying

Hope this helps

Krishna

Reference

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{ 22 comments… read them below or add one }

Ramya November 4, 2012 at 9:10 pm

what kind of method did you use to find the bit error rate?
how does it vary if I am supposed find out BER by using monte-carlo iterations?

Reply

Krishna Sankar November 12, 2012 at 7:06 am

@Ramya: For find the bit error rate, compare the received and transmitted bits and count the number of errors. Number of errors by total number of bits is the error rate

Reply

Siddharth May 12, 2012 at 9:54 pm

I am trying to code 16-FSK modulator and then compute average energy.How can I do this? Your help will be appreciated.

Reply

Krishna Sankar May 15, 2012 at 5:54 am

@Siddharth: Define your signal vector and find the mean square power.

Reply

zardosht February 26, 2012 at 12:56 pm

Hi krishna
I cant understand number of symbole, where is
k=log2M and for mfsk please help me.
i want some information about it.
thank you for your attention.

Reply

Krishna Sankar March 5, 2012 at 5:33 am

@zardosht: Hmm… the k = log2(M) means, if we have M constellation points we can send k = log2(M) bits in a constellation symbol. For eg, with 16QAM we can send 4 bits together.

Reply

Mahadevan Srinivasan December 30, 2009 at 7:20 am

Hi Krishna,

I recently found this website. Great Effort. .

Reply

Chandrakanth November 30, 2009 at 6:46 am

Can someone help me out on this please. I am looking for MAT LAB PROGRAM TO IMPLEMENT HUFFMAN CODING.

Thank you

Reply

Krishna Sankar December 7, 2009 at 4:54 am

@Chandrakanth: Sorry, I have not discussed Huffman coding till date.

Reply

Srinivas June 27, 2009 at 12:51 pm

Can someone help me out on this please.
For ISI channel with L real coeffs and BPSK modulation write a Viterbi algorithm decoder with decoding delay >0
Using Matlab
Plot BER vs SNR for varying delays

I need the matlab code.
Thank You

Reply

Krishna Sankar June 30, 2009 at 5:06 am

@Srinivas: I have not done MLSE equalization in multipath channel, but plan to do that in near future.

Reply

RAO June 16, 2009 at 6:23 am

hi krishna,

i want to plot the BER vs Eb/No of fsk coherent and noncoherent on awgn and raleigh channels.
can u help me in this regard?

reply

Reply

Krishna Sankar June 20, 2009 at 10:51 am

@RAO: Sorry, I have not tried modeling FSK with non-coherent demodulation OR in Rayleigh channel.

Reply

Joe June 8, 2009 at 6:08 am

What will BER be for M-FSK, if M=4?

Reply

Krishna Sankar June 9, 2009 at 4:57 am

@Joe: Am not sure about the precise answer, however a quick googling threw up the following link.
Noise Performances of different Modulation Schemes – Chris Haji-Michael
http://www.sunshadow.co.uk/mcad/BER%20Curves.pdf
Hope this is useful.

Reply

kalfika May 7, 2009 at 1:10 pm

Hi Krishna.
Your simulation is very useful for me.
Krishna, I want to ask about continuous fsk. Is continuous phase FSK same with fsk? How we define continuous phase in fsk?

Reply

Krishna Sankar May 12, 2009 at 5:21 am

@kalfika: Thanks.
Continuous phase FSK is not same as classical FSK. I think in literature the continuous phase variant of FSK is also known as MSK. I had discussed briefly about MSK transmission @ http://www.dsplog.com/2008/01/19/simulating-minimum-shift-keying-transmitter/

Hope this helps.

Reply

kalfika May 12, 2009 at 9:18 am

Can we use MSK to modulate 4-ary or 8-ary? Because I think MSK is just to modulate binary. What do you think about that?

Reply

Krishna Sankar May 20, 2009 at 5:06 am

@kalfika: Well, am also not aware of MSK systems using 4-ary/8-ary. Quick googling did not fetch me much results.

Reply

mohit April 11, 2009 at 11:36 am

hii

Reply

Krishna Sankar November 3, 2008 at 8:51 am

@davis: Well, given that we need to send both 1Hz and 2Hz, I do not think that the data bandwidth becomes equal to the carrier separation. Given the above understanding, am unable to understand your perspective. Can you please provide more details.

Reply

davis October 30, 2008 at 10:54 pm

Hi Krishna
i see that in the code the freqency for 0 is 1 Hz
and for 1 2 Hz
and that bit period is 1sec witch means that the data band width is the same as the seperation between the to carriers
How is this possible?
Thanks

Reply

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