Transcript Bandwidth and noise

Bandwidth and noise
Bandwidth and noise
• Bandwidth basically means how fast your
signal can change or how fast can you send
out symbols.
– Symbol is something you send out to represent
bit(s)
• Noise means that although you sent 1 to me, I
may receive something like 1+x, where x is the
noise added by the media.
Ideal case
• If the bandwidth is infinite and absolutely no
noise, how fast can you send/receive data?
Bandwidth
• If the media is of infinite bandwidth but with
some noise, how fast can you send/receive
data? Assuming that your device is fast
enough.
Noise
• If there is absolutely no noise but the
bandwidth is limited, how fast can you
send/receive data? Assuming that your device
is fine enough to tell the slightest differences
of signal voltage.
Shannon’s Theorem
• C=B*log(1+S/N)
– C is the capacity of the channel, B is the
bandwidth of the channel, S is power of the signal
and N is the power of the noise
– Channel capacity means how many bits you can
send out per second reliably
Shannon’s Theorem
• There is actually a very simple way to
understand Shannon’s theorem
– B means how fast can you send out symbols
– S/N determines how many bits each symbol
carries – why there is a log?
Limited Bandwidth
• Bandwidth is limited because of many reasons
– The wire itself, if too long, is a capacitor and slows
down voltage transition
– In wireless transmissions, the whole spectrum
shared by many communication parties and each
can have only a limited chunk of it
Limited bandwidth
• The trouble is, we live in a world with limited
bandwidth and some noise.
• Noise is easy to model.
• In mathematical languages, any signal can be
viewed as the sum of a series of sine waves on
different frequencies
• You pass the signal to a channel that can pass
frequencies up to B, all sine waves on frequencies
higher than B will be lost – you will receive a
distorted signal
Nyquist Theorem
• If the bandwidth is limited to B, in the ideal case when
there is no noise, how fast can you send/receive symbols?
– Note that the channel capacity is infinity because each symbol
can carry infinite number of bits
• Nyquist Theorem says that it only makes sense for you to
send/receive symbols at a speed of 2B – if B is 4KHz, you
send/receive 8K symbols per second – the baud rate is 8K
per second.
• Why? If a signal is band-limited by BHz, by taking 2B
samples per second, you can completely reconstruct it.
Nothing more can be reconstructed, so no point of sending.
How modulation is done
• Given data bit streams, translate them into
baseband waveforms
• Up-convert it to the carrier frequency
How demodulation is done
• Given the received waveform, down-convert it
to the baseband waveform
• Translate the baseband waveform to bit
streams
Wireless communications
• FDMA – Frequency division multiplexing
• TDMA – Time division multiplexing
• CDMA – Code division multiplexing
Wireless Demodulation
• You multiply the received signal with a sine
wave.
• Then you pass it to a low pass filter. This is the
baseband signal.
• Then you do phase tracking.
• Then you sample the wave, and get samples
of the data.
• Then you figure out what the bits are.
Demodulation
• With noise, it’s all about guessing, because you
don’t know what the noise is when this symbol is
sent as noise is random.
• You may know some statistics of the noise, based
on which you make your best guess.
• For example, let’s say 0 is 0 volt 1 is 5 volts.
Suppose you know that very rarely the noise
exceeds 2.5 volts. If you received a 2.2 volts, you
would guess it to be 0 or 1? What is the chance
that you got it right/wrong?
Maximum Likelihood Detection
• Detection – given a received signal, determine
which of the possible original signals was sent.
There are finite number of possible original
signals (2 for the binary case – 0 or 1)
• Compute a likelihood value for every possible
input, choose the one with largest likelihood –
maximum likelihood detection
Maximum Likelihood Detection
• There are two inputs, x1 and x2. Noise is n.
What you receive is y.
• If I sent x1, you receive y=x1 + n. If I sent x2,
you receive y=x2+n. You don’t know what I
sent and how large n is.
• You compute the likelihood of receiving y if I
sent xi, Li (i=1,2). If L1 > L2, you say I sent x1.
Else you say I sent x2.
• How to compute L1 and L2?
Maximum Likelihood Detection
• If n=0 always, y=x1 if I sent x1 and y=x2 if I sent x2. Of course x1 !=
x2. Will you make mistake in this case? What is the likelihood of
y=x2 if I sent x1?
• If n is not always zero, we assume n follows some probability
distribution. If it is Gaussian, the channel is called AWGN.
• Given y, the likelihood of x1 being sent is the likelihood that n=y-x1.
Similarly, the likelihood of x2 being sent is the likelihood that n=yx2. (likelihood is derived from probability, but likelihood could be
taking some values that probability cannot take depending on how
you define likelihood)
• So what you are doing is to compare the likelihood of n=y-x1 and
n=y-x2. So the detection rule is if p(n=y-x1)/p(n=y-x2) > 1, output
x1, else output x2.
• That’s all!
• Wait, what if you know that x1 is more likely to be sent than x2?
Wired Communication – Telephone
Company
• Dial-up – 56kbps
• DSL – Digital Subscriber Line
– ADSL: Asymmetric DSL, different upload and
download bandwidth
– Available bandwidth is about 1.1MHz, divided into
256 channels, one for voice, some unused or for
control, the rest divided among upstream and
downstream data. My DSL at Pittsburgh was 100kbps
upstream and 768kbps downstream
– How ADSL is set up. Fig. 2-29. The ADSL modem is 250
QAM modems operating at different frequencies. The
actual QAM depends on the noise.
Wired Communications – The Cable TV
Company
• Cable frequency allocation. Fig. 2-48.
– Downstream channel bandwidth is 6MHz. If using
QAM-64, how much of a speed we can get?
– Upstream channel is worse so use QAM-4.
– Upstream – stations contend for access (MAC
layer issue, will be discusses later)
– Downstream – no contention, from the head end
to user
– Shared medium, so some security is needed
Wired communication – Optical
Backbone
• SONET