In this study, we propose a wideband index modulation (IM) based on circularly -shifted chirps. To derive the proposed method, we first prove that a Golay complementary pair (GCP) can be constructed by linearly combining the Fourier series of chirps. We show that Fresnel integrals and/or Bessel functions, arising from sinusoidal and linear chirps, respectively, can lead to GCPs. We then exploit discrete Fourier transform-spread orthogonal frequency division multiplexing (DFT-s-OFDM) to obtain a low-complexity transmitter and receiver. We also discuss its generalization for achieving a trade-off between peak-to-mean envelope power ratio (PMEPR) and spectral efficiency (SE). Through comprehensive simulations, we compare the proposed scheme with DFT-s-OFDM with IM, orthogonal frequency division multiplexing (OFDM) with IM and complementary sequences (CSs) from Reed-Muller (RM) code. Our numerical results show that the proposed method limits the PMEPR while exploiting the frequency selectivity in fading channels without an auxiliary method.