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Understanding Logarithms and Channel Capacity in Signal Processing

Review logarithms and intermodulation noise in signal processing, including laws and errors. Explore crosstalk, impulse noise, channel capacity, and ADC's in a detailed breakdown.

elliotthernandez
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Understanding Logarithms and Channel Capacity in Signal Processing

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Presentation Transcript

  1. Logarithms • Log Review

  2. Logarithms • For example

  3. Logarithms

  4. Logarithms • Laws of Logarithms

  5. Intermodulation noise • results when signals at different frequencies share the same transmission medium

  6. the effect is to create harmonic interface at

  7. cause • transmitter, receiver of intervening transmission system nonlinearity

  8. Crosstalk • an unwanted coupling between signal paths. i.e hearing another conversation on the phone • Cause • electrical coupling

  9. Impluse noise • spikes, irregular pulses • Cause • lightning can severely alter data

  10. Channel Capacity • Channel Capacity • transmission data rate of a channel (bps) • Bandwidth • bandwidth of the transmitted signal (Hz) • Noise • average noise over the channel • Error rate • symbol alteration rate. i.e. 1-> 0

  11. Channel Capacity • if channel is noise free and of bandwidth W, then maximum rate of signal transmission is 2W • This is due to intersymbol interface

  12. Channel Capacity • Example w=3100 Hz C=capacity of the channel c=2W=6200 bps (for binary transmission) m = # of discrete symbols

  13. Channel Capacity • doubling bandwidth doubles the data rate if m=8

  14. Channel Capacity • doubling the number of bits per symbol also doubles the data rate (assuming an error free channel) (S/N):-signal to noise ratio

  15. Hartley-Shannon Law • Due to information theory developed by C.E. Shannon (1948) C:- max channel capacity in bits/second w:= channel bandwidth in Hz

  16. Hartley-Shannon Law • Example W=3,100 Hz for voice grade telco lines S/N = 30 dB (typically) 30 dB =

  17. Hartley-Shannon Law

  18. Hartley-Shannon Law • Represents the theoretical maximum that can be achieved • They assume that we have AWGN on a channel

  19. Hartley-Shannon Law C/W = efficiency of channel utilization bps/Hz Let R= bit rate of transmission 1 watt = 1 J / sec =enengy per bit in a signal

  20. Hartley-Shannon Law S = signal power (watts)

  21. Hartley-Shannon Law k=boltzman’s constant

  22. Hartley-Shannon Law assuming R=W=bandwidth in Hz In Decibel Notation:

  23. Hartley-Shannon Law S=signal power R= transmission rate and -10logk=228.6 So, bit rate error (BER) for digital data is a decreasing function of For a given , S must increase if R increases

  24. Hartley-Shannon Law • Example For binary phase-shift keying =8.4 dB is needed for a bit error rate of let T= k = noise temperature = C, R=2400 bps &

  25. Hartley-Shannon Law • Find S S=-161.8 dbw

  26. ADC’s • typically are related at a convention rate, the number of bits (n) and an accuracy (+- flsb) • for example • an 8 bit adc may be related to +- 1/2 lsb • In general an n bit ADC is related to +- 1/2 lsb

  27. ADC’s • The SNR in (dB) is therefore where about

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