Sound signals from the respiratory system are largely the harbingers of human health. Early diagnosis of respiratory tract diseases is of great importance as it creates irreversible effects on human health when delayed. This diagnostic in the medical world has been made possible thanks to machine learning and signal processing analysis. The coronavirus epidemic, which is in question today and deeply shakes the whole world, has been revealed the importance of this issue even more. In terms of the coronavirus pandemic, it has become the focus of researchers to differentiate symptoms from similar diseases such as normal flu or influenza. Among these symptoms, the difference in cough sound has played a distinctive role in the proposed study. Several pioneering studies have proven that almost two-thirds of people who get corona have a dry cough. At this stage, the information of studies based on cough constitutes the main framework of our study. On the other hand, the basis of this study is based on machine learning algorithms. Clinical data collected under the supervision of doctors in a reliable environment was used as dataset. This dataset consists of 16 subjects suspected of the coronavirus with a specific patient demographic. In this study, using the polymerase chain reaction (PCR) test, suspected subjects were divided into two groups as negative and positive. The negative and positive labels represent the patient with non-COVID and with a COVID-19 cough respectively. Using the 3D plot or waterfall representation of the signal frequency spectrum, the salient features of the cough data are revealed. In this way, COVID-19 can be differentiated from other coughs by applying effective feature extraction and classification techniques. Power Spectral Density (PSD) based on Short Time Fourier Transform (STFT) and Mel Frequency Cepstral Coefficients (MFCC) were chosen as the efficient feature extraction method. Finally, among the classification techniques the Support Vector Machine (SVM) algorithm, was applied to the processed signals in order to identify and classify COVID-19 cough. In terms of results evaluation, the cough of subjects with COVID-19 has obtained with 95.86% classification accuracy thanks to the RBF kernel function of SVM and the MFCC method. In other words, the diagnosis of COVID-19 coughs was obtained with 98.6% and 91.7% sensitivity and specificity measures respectively.