2.4 Modulated Signals 31Thus, the ratio of Es to the PSD of the noise No is given by ×==×sssoES TS BWNNNfBW (2.89)Thus, the ratio of the energy per symbol to the noise density is equal to the signal-to-noise ratio of the radio if the radio channel bandwidth is equal to the sym-bol frequency. This result is important because it will allow us to relate concepts in digital modulation to the SNR requirement of the radio.It is important to note that in digital communications it is often more common to see Eb, the energy per bit even when symbols are used. Eb is related to Es by: 2logsbEEM= (2.90)where log2M is the number of bits transmitted per symbol. Equation (2.89) can be rewritten as: =××21logbosES BWNNfM (2.91)Having discussed data and the concept of symbols, the next sections will de-scribe various modulation types. 2.4.1 Phase ModulationIn order to transmit information, more than just a carrier tone is needed. The tone has to change in some way over time to indicate what information is being sent. In general, a carrier has two properties that can be changed or modulated in order to convey information: amplitude and phase. We note that frequency modulation is Figure 2.12 Power spectral density of a data stream at baseband.