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The transceiver operations in the delay-Doppler (DD) domain in Zak-OTFSmodulation, including DD domain filtering at the transmitter and receiver,involve twisted convolution operation. The twisted convolution operations giverise to multiple integrals in the end-to-end DD domain input-output (I/O)relation. The I/O relation plays a crucial role in performance evaluation andalgorithm development for transceiver implementation. In this paper, we derivediscrete DD domain closed-form expressions for the I/O relation and noisecovariance in Zak-OTFS. We derive these expressions for sinc and Gaussian pulseshaping DD filters at the transmitter (Tx). On the receiver (Rx) side, threetypes of DD filters are considered, viz., $(i)$ Rx filter identical to Txfilter (referred to as `identical filtering'), $(ii)$ Rx filter matched to theTx filter (referred to as `matched filtering'), and $(iii)$ Rx filter matchedto both Tx filter and channel response (referred to as `channel matchedfiltering'). For all the above cases, except for the case of sinc identicalfiltering, we derive exact I/O relation and noise covariance expressions inclosed-form. For the sinc identical filtering case, we derive approximateclosed-form expressions which are shown to be accurate. Using the derivedclosed-form expressions, we evaluate the bit error performance of Zak-OTFS fordifferent Tx/Rx filter configurations. Our results using Vehicular-A (Veh-A)channel model with fractional DDs show that, while matched filtering achievesslightly better or almost same performance as identical filtering, channelmatched filtering achieves the best performance among the three.

Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters