In a frequency division duplex (FDD) massive MIMO system, downlink channel estimation poses several challenges with limited training duration being one impediment. Our previous work developed an algorithm to learn a dictionary in which the channel can be sparsely represented, and then leveraged compressed sensing framework to estimate the downlink channel. In this work, we extend the sparse channel representation framework to joint uplink and downlink channel modeling exploiting the similar scattering environment experienced by the signal during uplink and downlink transmission. We develop a joint dictionary learning algorithm in which joint sparse pattern of the uplink and downlink channels is enforced. This structure is utilized to improve downlink channel estimation from uplink training, which usually has much less training overhead in massive MIMO systems. Experimental results show that the proposed joint channel estimation improves the mean squared error (MSE) compared to downlink estimation only.