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Emerging wireless technologies, such as 5G and beyond, are bringing new use cases to the forefront, one of the most prominent being machine learning empowered health care. One of the notable modern medical concerns that impose an immense worldwide health burden are respiratory infections. Since cough is an essential symptom of many respiratory infections, an automated system to screen for respiratory diseases based on raw cough data would have a multitude of beneficial research and medical applications. In literature, machine learning has already been successfully used to detect cough events in controlled environments. In this paper, we present a low complexity, automated recognition and diagnostic tool for screening respiratory infections that utilizes Convolutional Neural Networks (CNNs) to detect cough within environment audio and diagnose three potential illnesses (i.e., bronchitis, bronchiolitis and pertussis) based on their unique cough audio features. Both proposed detection and diagnosis models achieve an accuracy of over 89%, while also remaining computationally efficient. Results show that the proposed system is successfully able to detect and separate cough events from background noise. Moreover, the proposed single diagnosis model is capable of distinguishing between different illnesses without the need of separate models.