We study joint estimation of the channel impulse response (CIR) and of the carrier frequency offset (CFO) for linear channels in which both the CIR and the noise statistics vary periodically in time. This model corresponds to interference-limited communications as well as to power line communication and doubly selective channels. We first consider the joint maximum likelihood estimator (JMLE) for the CIR and the CFO and show it has a high computational complexity and relatively low spectral efficiency. This motivates the derivation of two estimation schemes with higher spectral efficiency and lower computational complexity compared to the JMLE, obtained by exploiting both the periodicity of the channel and the fact that, typically, the delay-Doppler spreading function of the CIR is approximately sparse, without requiring a-priori knowledge of the sparsity pattern. The proposed estimation schemes are numerically tested and the results demonstrate that substantial benefits can be obtained by properly accounting for the approximate sparsity and periodicity in the design of the estimation scheme.