2. Fundamentals of Digital Communication

3. 2 Digital communication system Low Pass Filter SamplerQuantizer Channel Encoder Line Encoder Pulse Shaping Filters Source Encoder Modulator Multiplexer Input Signal Analog/ Digital To Channel Detector Receiver Filter De- Modulator From Channel Channel Decoder Digital-to-Analog Converter De- Multiplexer Signal at the user end Carrier Carrier Ref.

4. 3 Noiseless Channels and Nyquist Theorem For a noiseless channel, Nyquist theorem gives the relationship between the channel bandwidth and maximum data rate that can be transmitted over this channel. Nyquist Theorem C: channel capacity (bps) B: RF bandwidth m: number of finite states in a symbol of transmitted signal Example: A noiseless channel with 3kHz bandwidth can only transmit a maximum of 6Kbps if the symbols are binary symbols.

5. 4 Nyquist minimum bandwidth requirement The theoretical minimum bandwidth needed for baseband transmission of Rs symbols per second is Rs/2 hertz ?

6. 5 Shannon’s Bound for noisy channels There is a fundamental upper bound on achievable bandwidth efficiency. Shannon’s theorem gives the relationship between the channel bandwidth and the maximum data rate that can be transmitted over a noisy channel. Shannon’s Theorem C: channel capacity (maximum data-rate) (bps) B or W : RF bandwidth S/N: signal-to-noise ratio (no unit)

7. 6 Shannon limit … Shannon theorem puts a limit on transmission data rate, not on error probability: Theoretically possible to transmit information at any rate R b, where R b  C with an arbitrary small error probability by using a sufficiently complicated coding scheme. For an information rate R b > C, it is not possible to find a code that can achieve an arbitrary small error probability.

8. 7 Shannon limit … C/W [bits/s/Hz] SNR [dB] Practical region Unattainable region

9. 8 Shannon limit … There exists a limiting value of below which there can be no error-free communication at any information rate. By increasing the bandwidth alone, the capacity cannot be increased to any desired value. Shannon limit

10. 9 Shannon limit … W/C [Hz/bits/s] Practical region Unattainable region -1.6 [dB]

11. 10 Bandwidth efficiency plane R<C Practical region R>C Unattainable region R/W [bits/s/Hz] Bandwidth limited Power limited R=C Shannon limit MPSK MQAM MFSK M=2 M=4 M=8 M=16 M=64 M=256 M=2M=4 M=8 M=16

12. 11 Error probability plane (example for coherent MPSK and MFSK) Bit error probability M-PSK M-FSK k=1,2 k=3 k=4 k=5 k=4 k=2 k=1 bandwidth-efficient power-efficient

13. 12 M-ary signaling Bandwidth efficiency: Assuming Nyquist (ideal rectangular) filtering at baseband, the required passband bandwidth is: M-PSK and M-QAM (bandwidth-limited systems) Bandwidth efficiency increases as M increases. MFSK (power-limited systems) Bandwidth efficiency decreases as M increases.

14. 13 Power and bandwidth limited systems Two major communication resources: Transmit power and channel bandwidth In many communication systems, one of these resources is more precious than the other. Hence, systems can be classified as: Power-limited systems: save power at the expense of bandwidth (for example by using coding schemes) Bandwidth-limited systems: save bandwidth at the expense of power (for example by using spectrally efficient modulation schemes)

15. 14 Goals in designing a DCS Goals: Maximizing the transmission bit rate Minimizing probability of bit error Minimizing the required power Minimizing required system bandwidth Maximizing system utilization Minimize system complexity

16. 15 Limitations in designing a DCS The Nyquist theoretical minimum bandwidth requirement The Shannon-Hartley capacity theorem (and the Shannon limit) Government regulations Technological limitations Other system requirements (e.g satellite orbits)

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