Theoretical BER vs SNR for binary ASK and FSK

 

Theoretical Ber vs SNR for Amplitude Shift Keying (ASK)

The theoretical bit error rate (BER) for binary Amplitude Shift Keying (ASK) as a function of the signal-to-noise ratio (SNR) can be derived using the following expression:

Where:

Q(x) is the Q-function, which is the tail probability of the standard normal distribution.

SNR is the signal-to-noise ratio.

N0 is the noise power spectral density.

For a binary ASK system in an additive white Gaussian noise (AWGN) channel, the SNR is usually expressed in terms of the energy per bit 𝐸𝑏 and the noise power spectral density 𝑁0 :

Where Eb/N0 is the SNR and in mathematics Q(x) = 0.5 * erfc(x/√2)

This expression shows that the BER decreases as the SNR increases, meaning that a higher SNR leads to better performance (lower BER) for binary ASK. 

Theoretical BER vs SNR for Frequency Shift Keying (FSK)

Formulae for bit error rate (BER) of binary FSK is 

Where Q is the Q function

In mathematics Q(x) = 0.5 * erfc(x/√2)

So, theoretical BER for binary FSK will be

Where:

Q(x) is the Q-function.

Eb is the energy per bit. 

N0 is the noise power spectral density.

erfc(x) is the complementary error function.

 

 Fig: Theoretical BER vs SNR for Binary ASK Modulation

 

  

  Fig: Theoretical BER vs SNR for Binary FSK Modulation

Similarities:

For both ASK and BFSK, the BER decreases as the SNR increases, indicating better performance at higher SNR values.

The formulas for BER in both cases involve the complementary error function, indicating that they follow similar trends, though the constants and scaling factors differ slightly.

 

MATLAB Code for theoretical BER vs SNR for BASK

% The code is written by SalimWireless.Com % Binary ASK Modulation: Theoretical BER Performance Analysis clc; clear all; close all; SNRdB = 0:20; % SNR in dB SNR = 10.^(SNRdB/10); % Convert SNR from dB to linear scale % Calculate theoretical BER for Binary ASK BER_th = (1/2) * erfc(0.5 * sqrt(SNR)); % Plot theoretical BER semilogy(SNRdB, BER_th, '-rh', 'linewidth', 2.5); grid on; title('Theoretical Bit Error Rate vs. SNR for Binary ASK Modulation'); xlabel('SNR (dB)'); ylabel('BER'); legend('Theoretical'); axis([0 20 1e-5 1]);

MATLAB Code for theoretical BER vs SNR for BFSK

% The code is written by SalimWireless.Com clc; clear; close all; % Parameters SNRdB = 0:1:10; % SNR in dB SNR = 10.^(SNRdB/10); % Convert SNR from dB to linear scale % Calculate theoretical BER for BFSK BER_th = (1/2) * erfc(sqrt(SNR / 2)); % Display theoretical BER vs SNR disp('SNR (dB) Theoretical BER'); disp([SNRdB', BER_th']); % Plot theoretical BER figure; semilogy(SNRdB, BER_th, '-kh', 'LineWidth', 2); xlabel('SNR (dB)'); ylabel('Bit Error Rate (BER)'); title('Theoretical BER vs SNR for BFSK'); grid on;

Also read about

[1] Theoretical BER vs SNR for binary PSK (BPSK)

[2] Constellation Diagrams of ASK, FSK, and PSK