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Shingled magnetic recording on bit patterned media with areal densities of 2-4 Tbit/in2 is examined. The use of a shingled recording scheme allows wider write poles and dots with higher anisotropy to be used. The down-track and cross-track write margins can be tailored to suit the tolerances of the drive by varying the bit cell aspect ratio. Readback waveforms are calculated for sensors of various size, including the effect of thermal noise in the sensor, and the combination of a wide sensor and bit aspect ratio (BAR) of 1:1 is found to give the highest SNR.