Quantum computing has the potential to revolutionize the signal-processing field by providing more efficient methods for analyzing signals. This thesis explores the application of quantum computing in signal analysis synthesis for compression applications. More specifically, the study focuses on two key approaches: quantum Fourier transform (QFT) and quantum linear prediction (QLP). The research is motivated by the potential advantages offered by quantum computing in massive signal processing tasks and presents novel quantum circuit designs for QFT, quantum autocorrelation, and QLP, enabling signal analysis synthesis using quantum algorithms. The two approaches are explained as follows. The Quantum Fourier transform (QFT) demonstrates the potential for improved speed in quantum computing compared to classical methods. This thesis focuses on quantum encoding of signals and designing quantum algorithms for signal analysis synthesis, and signal compression using QFTs. Comparative studies are conducted to evaluate quantum computations for Fourier transform applications, considering Signal-to-Noise-Ratio results. The effects of qubit precision and quantum noise are also analyzed. The QFT algorithm is also developed in the J-DSP simulation environment, providing hands-on laboratory experiences for signal-processing students. User-friendly simulation programs on QFT-based signal analysis synthesis using peak picking, and perceptual selection using psychoacoustics in the J-DSP are developed. Further, this research is extended to analyze the autocorrelation of the signal using QFTs and develop a quantum linear prediction (QLP) algorithm for speech processing applications. QFTs and IQFTs are used to compute the quantum autocorrelation of the signal, and the HHL algorithm is modified and used to compute the solutions of the linear equations using quantum computing. The performance of the QLP algorithm is evaluated for system identification, spectral estimation, and speech analysis synthesis, and comparisons are performed for QLP and CLP results. The results demonstrate the following: effective quantum circuits for accurate QFT-based speech analysis synthesis, evaluation of performance with quantum noise, design of accurate quantum autocorrelation, and development of a modified HHL algorithm for efficient QLP. Overall, this thesis contributes to the research on quantum computing for signal processing applications and provides a foundation for further exploration of quantum algorithms for signal analysis synthesis.