Phonocardigraphy (PCG) is the graphical representation of heart sounds. The PCG signal contains useful information about the functionality and the condition of the heart. It also provides an early indication of potential cardiac abnormalities. Extracting cardiac information from heart sounds and detecting abnormal heart sounds to diagnose heart diseases using the PCG signal can play a vital role in remote patient monitoring. In this paper, we have combined different signal processing techniques and a deep learning method to denoise, compress, segment, and classify PCG signals effectively and accurately. First, the PCG signal is denoised and compressed by using a multi-resolution analysis based on the Discrete Wavelet Transform (DWT). Then, a segmentation algorithm, based on the Shannon energy envelope and zerocrossing, is applied to segment the PCG signal into four major parts: the first heart sound (S1), the systole interval, the second heart sound (S2), and the diastole interval. Finally, Mel-scaled power spectrogram and Mel-frequency cepstral coefficients (MFCC) are employed to extract informative features from the PCG signal, which are then fed into a classifier to classify each PCG signal into a normal or an abnormal signal by using a deep learning approach. For the classification, a 5-layer feed-forward Deep Neural Network (DNN) model is used, and overall testing accuracy of around 97.10% is achieved. Besides providing valuable information regarding heart condition, this signal processing approach can help cardiologists take appropriate and reliable steps toward diagnosis if any cardiovascular disorder is found in the initial stage.