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This paper addresses the waveform design problem to estimate the target impulse response (TIR) of the temporally correlated extended target in a cognitive radar, subject to a detection constraint and a peak-to-average power ratio constraint. Owing to these types of constraints and the convolution operation of the waveform in the time domain, the formulated optimization problem for minimizing the mean square error of the estimated TIR based on Kalman filtering is a complex non-convex problem. To this end, an auxiliary variable is first introduced to modify the original problem, and the non-convex problem is converted to a convex problem with respect to the matrix variable. Then, a trick is used for replacing the matrix variable with the vector variable by utilizing the properties of the Toeplitz matrix. Moreover, the convex problem is further decomposed into three simple sub-problems which can be solved efficiently. Finally, the optimal waveform can be obtained efficiently through cognitive iteration combined with the nearest neighbor method. The simulation results illustrate that the proposed method is superior to the existing method in terms of the estimation performance and computational complexity when designing the constrained waveform.