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In broadband networks: −→ use analog signals to carry digital data. Page 2. CS 422. Park. Important task: analog data is often digitized. −→ useful: why? − ...
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Digital vs. Analog Data
Digital data: bits.
−→ discrete signal −→ both in time and amplitude
Analog “data”: audio/voice, video/image
−→ continuous signal −→ both in time and amplitude
Both forms used in today’s network environment.
−→ burning CDs −→ audio/video playback
In broadband networks:
−→ use analog signals to carry digital data
Important task: analog data is often digitized
−→ useful: why? −→ it’s convenient −→ use full power of digital computers −→ simple form: digital signal processing −→ analog computers are not as versatile/programmable −→ cf. “Computer and the Brain,” von Neumann (1958)
How to digitize such that digital representation is faithful?
−→ sampling −→ interface between analog & digital world
Sampling criterion for guaranteed faithfulness:
Sampling Theorem (Nyquist): Given continuous bandlimited signal s(t) with S(ω) = 0 for |ω| > W , s(t) can be reconstructed from its samples if
ν > 2 W
where ν is the sampling rate.
−→ ν: samples per second
Remember simple rule: sample twice the bandwidth
Issue of digitizing amplitude/magnitude ignored
−→ problem of quantization −→ possible source of information loss −→ exploit limitations of human perception −→ logarithmic scale
Compression
Information transmission over noiseless medium
−→ medium or “channel” −→ fancy name for copper wire, fiber, air/space
Sender wants to communicate information to receiver over noiseless channel.
−→ can receive exactly what is sent −→ idealized scenario
Part II. Compression machinery:
Ex.: Σ = {A, C, G, T }; need at least two bits
−→ pros & cons?
Note: code book F is not unique
−→ find a “good” code book −→ when is a code book good?
A fundamental result on what is achievable to attain small L.
−→ kind of like speed-of-light
First, define entropy H of source 〈Σ, p〉
H =
a∈Σ
p (^) a log
p (^) a
Ex.: Σ = {A, C, G, T }; H is maximum if pA = p (^) C = p (^) G = p (^) T = 1/4.
−→ when is it minimum?
Source Coding Theorem (Shannon): For all code books F , H ≤ LF
where LF is the average code length under F.
Furthermore, LF can be made to approach H by selecting better and better F.
Remark:
Would like: if received code word w = wc for some symbol c ∈ Σ, then probability that actual symbol sent is indeed c is high
−→ Pr{actual symbol sent = c | w = wc} ≈ 1 −→ noiseless channel: special case (prob = 1)
In practice, w may not match any legal code word:
−→ for all c ∈ Σ, w 6 = wc −→ good or bad? −→ what’s next?
Shannon showed that there is a fundamental limitation to reliable data transmission.
→ the noisier the channel, the smaller the reliable throughput → overhead spent dealing with bit flips
Definition of channel capacity C: maximum achievable reliable data transmission rate (bps) over a noisy channel (dB) with bandwidth W (Hz).
Channel Coding Theorem (Shannon): Given band- width W , signal power PS, noise power PN , channel sub- ject to white noise,
C = W log
bps.
P (^) S /PN : signal-to-noise ratio (SNR)
−→ upper bound achieved by using longer codes −→ detailed set-up/conditions omitted
Signal-to-noise ratio (SNR) is expressed as
dB = 10 log 10 (P (^) S /PN ).
Example: Assuming a decibel level of 30, what is the channel capacity of a telephone line?
Answer : First, W = 3000 Hz, PS /PN = 1000. Using Channel Coding Theorem,
C = 3000 log 1001 ≈ 30 kbps.
−→ compare against 28.8 kbps modems −→ what about 56 kbps modems? −→ DSL lines?
Digital vs. Analog Transmission
Two forms of transmission:
Four possibilities to consider:
Delay distortion: different frequency components travel at different speeds.
Most problematic: effect of noise
−→ thermal, interference,...
Analog Transmission of Digital Data
Three pieces of information to manipulate: amplitude, frequency, phase.
0 1 1 0 0 1
