Although the design of hybrid precoders and combiners separately from the complete channel state information (CSI) offers satisfactory performance, the resulting spatial multiplexing channel may not always be orthogonal during communication. Also, acquiring CSI to design optimal precoders and combiners poses several challenges, particularly in millimeter wave (mmWave) channel estimation, and getting the sensing matrix is equivalent to designing the precoders and combiners. For this, we propose a new iterative method based on alternating minimization to design the optimal sensing matrix (incoherent projection matrix) with the given dictionary to minimize the mutual coherence values (µ mx , µ ave and µ all) simultaneously according to the equiangular tight frame (ETF) properties for achieving better-compressed sensing (CS) recovery performance. Then, in order to derive the best hybrid precoders and combiners jointly from the optimally designed sensing matrix, we formulate the optimization design problem as the nearest Kronecker product (NKP) problem. The proposed sensing matrix design works better at lowering the mutual coherence values concurrently with the straightforward shrinkage function, according to simulation findings of mutual coherence values evolution versus outer iteration numbers. In comparison to existing codebookbased hybrid precoder/combiner schemes, the proposed joint hybrid precoder and combiner design improves the performance of the simulation results obtained by multi-stage CS-based mmWave channel estimation in terms of channel estimation accuracy and spectral efficiency (SE). INDEX TERMS Millimeter-wave channel estimation, Multi-stage CS approach, Hybrid mmWave MIMO transceiver, Joint hybrid precoder and combiner design, Equiangular tight frame, Mutual coherence values, Incoherent projection matrix.