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The widespread usage of the Radio Frequency (RF) spectrum for wireless and mobile communication systems generated a significant spectrum scarcity. The Joint Radar-Communication System (JRCS) provides a framework to simultaneously utilize the allocated radar spectrum for sensing and communication purposes. Generally, a Successive Interference Cancellation (SIC) based receiver is applied to mitigate mutual interference in the JRCS configuration. However, this SIC receiver model introduces a communication residual component. In response to this issue, the article presents a novel measurement model based on communication residual components for various radar waveforms. The radar system's performance within the JRCS framework is then evaluated using the Fisher Information Matrix (FIM). The radar waveforms considered in this investigation are rectangular pulse, triangular pulse, Gaussian pulse, Linear Frequency Modulated (LFM) pulse, LFM-Gaussian pulse, and Non-Linear Frequency Modulated (NLFM) pulse. After that, the Kalman filter is deployed to estimate the target kinematics (range and range rate) of a single linearly moving target for different waveforms. Additionally, range and range rate estimation errors are quantified using the Root Mean Square Error (RMSE) metric. Furthermore, the Posterior Cramer-Rao Lower Bound (PCRLB) is derived to validate the estimation accuracy of various waveforms. The simulation results show that the range and range rate estimation errors are within the PCRLB limit at all time instants for all the designated waveforms. The results further reveal that the NLFM pulse waveform provides improved range and range rate error performance compared to all other waveforms.