Which one of the following digital modulation schemes has the bit error rate as \(\dfrac{1}{2}\rm{erfc} \left( \sqrt{\dfrac{E_b}{N_0}} \right)\) ?

option 1 is correct

BPSK

S₁(t)= A_c cos (2πf_c t)

S₂(t)= -A_c cos (2πf_c t)

Bit error rate (BER) for BPSK given as 

\((BER)_{BPSK}= Q\sqrt \frac{A_C^2\times T_b}{N_o }\)

noise power =N_o/2

bit energy(E_b)=A_c²T_b/2

\((BER)_{BPSK}= Q\sqrt \frac{2\times E_b}{N_o }\)

Relation between Q(x) and erfc are 

Q(x)=1/2 erfc (x/√ 2)

\((BER)_{BPSK}=Q\sqrt { (2E_b/N_o)}=1/2× erfc(\sqrt {(E_b/N_o)}\)