Derivation and simulation of the bit and symbol error rate probabilities of a QFSK system with non-coherent detection.

1.  
Tom Cruz
05/02/2017

2. 
Part I

3. 𝑠𝑖 𝑡 =
2𝐸
𝑇
cos(𝜔𝑖 𝑡 + 𝜎) for 0 ≤ t ≤ T , i = 1,…,M
𝑃𝐵 =
1
2
𝑃 𝐻2 𝑠1) +
1
2
𝑃 𝐻1 𝑠2)
=
1
2 −∞
0
𝑝(𝑧| 𝑠1) 𝑑𝑧 +
1
2 0
∞
𝑝(𝑧| 𝑠2) 𝑑𝑧
𝑝 𝑧 𝑠1 = 𝑝(−𝑧|𝑠2) 𝑃𝐵 =
0
∞
𝑝(𝑧| 𝑠2) 𝑑𝑧 𝑃𝐵 = 𝑃 𝑧1 > 𝑧2 𝑠2)
A signal set for Noncoherently Detected Orthogonal FSK is defined as:
Probability of an error is the sum of the
probabilities of all the ways that an error can
occur, represented by:
Setting the decision threshold
to γ = 0, through symmetry we
get
Reducing the integral to Which can be written as
where z1 and z2 denote the outputs from the envelope
The input to the detector
is the received signal:
where n(t) is a white
Gaussian noise process
𝑟 𝑡 = 𝑠𝑖 𝑡 + 𝑛(𝑡)

4. 𝑝 𝑧1 𝑠2 =
𝑧1
𝜎0
2 exp −
𝑧1
2
2𝜎0
2 ; 𝑧1 ≥ 0
0 ; 𝑧1 < 0
𝑝 𝑧2 𝑠2 =
𝑧2
𝜎0
2 exp −
𝑧2
2
+ 𝐴2
2𝜎0
2 𝐼0
𝑧2 𝐴
𝜎0
2
0 ; 𝑧2 < 0
; 𝑧2
≥ 0
𝐴 = 2𝐸
𝑇
𝐼0 𝑥 =
1
2𝜋 0
2𝜋
exp 𝑥 cos 𝜃 𝑑𝜃
In the case where s2 is sent with a Gaussian noise random variable only, no
signal component will be present.
A Gaussian distribution into the nonlinear envelope detector yields a Rayleigh
distribution:
where σ^2 is the noise at
the filter output
Since the other input to the lower envelope detector is a sinusoid with noise,
the pdf is:
Io, known as the modified zero-order Bessel function of the first kind, is defined
as:
where

5. 𝑃𝐵 = 𝑃 𝑧1 > 𝑧2 𝑠2)
𝑃𝐵 =
0
∞
𝑝(𝑧2| 𝑠2)
0
∞
𝑝(𝑧1| 𝑠2) 𝑑𝑧1 𝑑𝑧2
𝑃𝐵 =
0
∞
𝑧2
𝜑0
2 exp −
𝑧2
2
+ 𝐴2
2𝜎0
2 𝐼0
𝑧2 𝐴
𝜑0
2
0
∞
𝑧1
𝜎0
2 exp −
𝑧1
2
2𝜎0
2 𝑑𝑧1 𝑑𝑧2
𝑃𝐵 =
1
2
𝑒𝑥𝑝 −
𝐴2
4𝜎0
2
𝜎0
2
= 2
𝑁0
2
𝑊𝑓
𝑃𝐵 =
1
2
𝑒𝑥𝑝 −
𝐴2
4𝑁0 𝑊𝑓
𝑃𝐵 =
1
2
𝑒𝑥𝑝 −
𝐴2
𝑇
4𝑁0
𝑊𝑓 =
1
𝑇
𝑷 𝑩 =
𝟏
𝟐
𝒆𝒙𝒑 −
𝑬 𝒃
𝟐𝑵 𝟎
When s2 is transmitted, the receiver makes an error whenever the envelope
sample obtained from the upper channel (due to noise) exceeds the envelope
sample obtained from the lower channel (due to signal plus noise).
Therefore the probability of this error can be obtained through integrating the
following:
The integral then evaluates
to:
The filter output noise is expressed
as:
Substituting in the filter output noise
gives:
=
where
Substituting Wf and A results in:

6. 𝑃𝐸 𝑀 =
1
𝑀
𝑒𝑥𝑝 −
𝐸𝑠
𝑁0
𝑗−2
𝑀
−1 𝑗
𝑀
𝑗
𝑒𝑥𝑝
𝐸𝑠
𝑗𝑁0
𝑀
𝑗
=
𝑀!
𝑗! 𝑀 − 𝑗 !
𝑃𝐵 =
1
2
𝑒𝑥𝑝 −
𝐸 𝑏
2𝑁0
Using the binomial coefficient producing the number of ways in which j symbols
out of M may be in error.
𝑷 𝑩
𝑷 𝑬
=
𝑴
𝟐
𝑴 − 𝟏
From our original derivation of the Bit Error rate Probability, we produced a result
where the probability of error is exponentially related to the energy to noise ratio
of the bits, written as:
Likewise, a relationship between the probability of symbol errors and the energy
to noise ratio of the symbols can be written as:
Therefore, the relationship between the bit and symbol error probability for an M-
ary FSK signal set is:

7. 
Part II

8. clear
M = 4; %Size of symbol set
N = 10^5; % number of bits
T = 8; % Symbol duration
t = [0:1/T:0.99]; % Sampling instants
tR = kron(ones(1,N),t); % Repeating the sampling instants
Eb_N0_dB = [0:2:10]; % Multiple Eb/N0
for ii = 1:length(Eb_N0_dB)
% Generating the bits
ip = rand(1,2*N)>0.5; % generating 0,1 with equal probability
ip_odd = ip(1:2:end);
ip_even = ip(2:2:end);
freqM = ip_odd*2+ip_even*1+1;
% converting the bits into frequency, bit00 -> frequency of 1,
bit01 -> frequency of 2, bit10 -> frequency of 3, bit11 -> frequency of 4
freqR = kron(freqM,ones(1,T)); % repeating
x = (sqrt(4)/sqrt(T))*cos(2*pi*freqR.*tR); %generating the FSK modulated signal
% Noise
n = 1/sqrt(2)*[randn(1,N*T)]; % white gaussian noise, 0dB variance
1. Initialize Parameters
2. Modulate Signal & Define Noise

9. % Non-Coherent receiver
y = x + 10^(-Eb_N0_dB(ii)/20)*n; % additive white Gaussian noise
op1s = conv(y, sqrt(2/T)*sin(2*pi*1*t)); % correlating with frequency 1
op1c = conv(y, sqrt(2/T)*cos(2*pi*1*t));
op2s = conv(y, sqrt(2/T)*sin(2*pi*2*t)); % correlating with frequency 2
op2c = conv(y, sqrt(2/T)*cos(2*pi*2*t));
op3s = conv(y, sqrt(2/T)*sin(2*pi*3*t)); % correlating with frequency 3
op3c = conv(y, sqrt(2/T)*cos(2*pi*3*t));
op4s = conv(y, sqrt(2/T)*sin(2*pi*4*t)); % correlating with frequency 4
op4c = conv(y, sqrt(2/T)*cos(2*pi*4*t));
% Squaring
op1 = abs(op1s(T+1:T:end).^2 + op1c(T+1:T:end).^2);
op2 = abs(op2s(T+1:T:end).^2 + op2c(T+1:T:end).^2);
op3 = abs(op3s(T+1:T:end).^2 + op3c(T+1:T:end).^2);
op4 = abs(op4s(T+1:T:end).^2 + op4c(T+1:T:end).^2);
3. Design Non-Coherent Receiver

10. % Demodulation
for i=1:N
op = [op1(i),op2(i),op3(i),op4(i)];% find which is largest of
4 frequencies and demodulate it
[m,in] = max(op);
ipHat(2*i-1) = fix((in-1)/2);
ipHat(2*i) = mod((in-1),2);
end
ipHat_odd = ipHat(1:2:end);
ipHat_even = ipHat(2:2:end);
freqMHat = ipHat_odd*2+ipHat_even*1+1; % The symbol after
demodulation
nErrb(ii) = size(find([ip - ipHat]),2); % counting the number of
errors of Bit
nErrs(ii) = size(find([freqM - freqMHat]),2); % counting the number
of errors of Symbol
end
4. Demodulation

11. 4. Calculate Simulation and Theoretical Values
simBer = nErrb/2/N;% non-coherent simulate BER (QFSK)
simSer = nErrs/N; % non-coherent simulate SER (QFSK)
theorySer_total = 0; %Initialization
for i = 2:M
theorySer_sum = ((-1)^i)*(factorial(M)/(factorial(M-
i)*factorial(i)))*exp(log2(M)*(10.^(Eb_N0_dB/10))/i);
theorySer_total = theorySer_total + theorySer_sum;
end
theorySer = 1/M*exp(-2*(10.^(Eb_N0_dB/10))).*theorySer_total; % non-coherent
theoretical SER (QFSK)
theoryBer = 0.5*(M/(M-1))*theorySer; % non-coherent theoretical BER (QFSK)
close all
figure
semilogy(Eb_N0_dB,theoryBer,'b-');
hold on
semilogy(Eb_N0_dB,simBer,'rx-');
grid on
legend('QFSK: Theoretical', 'QFSK:
Simulation');
xlabel('Eb/No, dB')
ylabel('Bit Error Rate')
title('Bit Error Probability Curve')
5. Graph Results
figure
semilogy(Eb_N0_dB,theorySer,'b-');
hold on
semilogy(Eb_N0_dB,simSer,'rx-');
grid on
legend('QFSK: Theoretical', 'QFSK:
Simulation');
xlabel('Eb/No, dB')
ylabel('Symbol Error Rate')
title('Symbol Error Probability Curve')

13. 
Supporting information on Non-Coherent M-ary FSK

14.  Advantages of FSK
o FSK is not very susceptible to noise, since the voltage spikes
caused by noise affects the carrier’s amplitude, but do not affect the
carrier’s frequency.
o FSK is ideally a constant envelope modulation; hence, more power-
efficient class-C non-linear Power Amplifiers can be used in the
transmitter
o FSK is more bandwidth efficient than ASK
 Disadvantages of FSK
o The difference between coherent and non-coherent FSK detection
is not significant for higher FSK levels
o Coherent FSK is not often used in practice due to the difficulty, and
cost, in generating two reference frequencies close together at the
receiver
• The noncoherent FSK demodulator is considerably easier to build since
coherent reference signals need to be generated.

15. Figure 1. Receiver for Binary FSK

16.  Atlanta RF – Link Budget Analysis: Digital Modulation
 Bernard Sklar – Digital Communications: Fundamentals
and Applications