Theoretical BER vs SNR for binary ASK and FSK

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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:

If we map the binary signals to 1 and -1 in ASK, the probability of bit error will be:

BER = Q(√(2*SNR))

 

If we map the binary signals to 0 and 1 in ASK, the probability of bit error will be:

 

 BER = Q(√(SNR/2))

 

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.

Where Q is the Q function

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

 

Calculate the Probability of Error using Q-function for ASK:

For ASK with amplitudes 0 and 1:

• When bit '0' is transmitted, the received signal is noise only.

• When bit '1' is transmitted, the received signal is 1 + noise.

• The receiver makes a decision at the threshold 0.5.

• If bit 0 is transmitted, noise must exceed +0.5.

If bit 1 is transmitted, noise must decrease the signal below 0.5.

In either case, the noise is Gaussian with mean = 0 and variance = N₀/2. The probability of noise exceeding ±0.5 can be calculated with the Q-function:

P_b = Q(0.5/σ)

Where:

σ = √(0.5/2)

So:

P_b = Q(0.5/√(N₀/2)) = Q(√(2(0.5)²/N₀))

Since:

SNR = (0.5)² / N₀

We get:

P_b = Q(√(SNR/2))

 

Theoretical BER vs SNR for Frequency Shift Keying (FSK)

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

BER = Q(√(SNR))

 

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. Theoretical and simulated BER vs. SNR for ASK, FSK, and PSK
3. Theoretical BER vs SNR for m-ary PSK and QAM
4. Constellation Diagrams of ASK, FSK, and PSK