A communication system has a much broader scope. For example it allows you to select whom to communicate with and in what form. Also it might contain scheduling systems or automations.
Digital signal processing is often part of a communication system. There are a few roles for DSP. First of all, the world is analog and data is digital. So for example if you want to transmit voice over the internet, it has to sample your voice and digitize that. The digitized signal has to be sent over internet communication lines to the receiver. Since the cables live in the real world, the digital signals have to be converted into power levels, possibly with modulation. Also you might want to apply certain filters to improve the audio quality.
Communication is the act of transmitting information (eg: “I am hungry”). As part of that process, the information is encoded and transmitted (eg: you speak words). The encoded transmission (eg: the sound of your voice) is called a signal. Digital signal processing is the specific act of encoding, decoding or transforming digital signals using digital logic.
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