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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:

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:

    Pb = Q(0.5/σ)

    Where:

    σ = √(0.5/2)

    So:

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

    Since:

    SNR = (0.5)2 / N0

    We get:

    Pb = 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



    MATLAB Code for theoretical BER vs SNR for BFSK



    Also read about
    [1] Theoretical BER vs SNR for binary PSK (BPSK)
    [2] Constellation Diagrams of ASK, FSK, and PSK

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