Transcript Week5_1

Review
• The very basic wireless communication system
Review
• Filters are used to translate the bits into baseband
waveforms. We use RRC filers.
• This process is called ``pulse shaping.’’
• Every symbol (in BPSK, each symbol is either -1 or 1)
will generate a waveform, which depends on the
*impulse response* of the low pass filter (in BPSK, it is
either just the impulse response or the impulse
response times -1).
• Remember the shape of the impulse response!
• The transmitter sends a symbol to the LPF every T
seconds. T is called the symbol time. In GNU SDR, T is
2us.
Review
• The signals we can actually send is the following,
where I(t) and Q(t) are the baseband waveforms.
In BPSK, the cosine is multiplied with a waveform,
the sine is not.
Review
• The baseband waveform is the addition of the
impulse response waveforms generated by the
data symbols.
– The impulse response is usually written as h(t).
– Let the data symbols sent to the low pass filter be
x[n] at time n. x[n] will add a voltage of x[n]h(tnT) to the baseband waveform at time t.
Review
• Multi-path. Signals travel infinite number of
paths to reach the receiver. The received
signal is the addition of signals from all paths:
where A is the attenuation, \tau is the delay.
• After received the signal, we will first multiply
it with sine and cosine wave, and pass it
through the LPF, as explained before.
Review
• Complex channels – because we can send both a sine and a
cosine wave, which can be conveniently represented as a
complex wave.
• The sender and receiver will try to use the same frequency,
but they cannot. Consider the cosine branch and consider a
single path.
• So, after the low pass filter, it becomes
• Similarly, the sine branch will produce
• Therefore, the cosine and sine branch can be regarded
as a complex number
Review
• To establish the communication, we will first have to
reproduce the baseband waveform. Which means that we
have to get rid of , a process called carrier phase
tracking.
• We take samples from the received baseband waveform to
get
• Assume the samples are taken at perfect time, i.e., when
the impulse response is at the peak, so the sample should
be either 1 or -1 in the BPSK case. The ideal sample is called
“constellation point.” Extract the phase error and adjust the
phase.
Review
• Timing synchronization – how to
take samples at the correct time.
• The basic Mueller and Muller
algorithm: u=a_{k-1} x_k – a_{k}
x_{k-1}.
• u is the timing error.
• In this example, if off by a delta,
x_{k-1} will be deducted by the
impulse response at T+delta,
which is negative, while x_{k} will
be added by the impulse response
at -T+delta, which is positive.
k-1
k
K+1
Review
• While
• The first line is the one that does the work. In
BPSK, it becomes
Review
• The current GNU SDR implementation uses
the optimized mm algorithm. Assume a
sample error of \delta. The sample at time n is
• The timing error is calculated as
• The reason is that
Review
• Dealing with multipath.
Review
• Important thing to remember is that even with
perfect carrier phase tracking and symbol timing,
when taking a sample, it will contain some thing
from the neighboring symbols.
• At time 0, the path with delay will contribute
to this sample, but for k!=0, also
• And there are infinite number of paths.
• In other words, if you take a sample, will be
something like
, where g[i] is the
summation of all paths.
Review
• The red
curve
represents
the other
path.
k-1
k
K+1
Review
• We can use equalization. The basic idea is to
subtract the contributions of non-relevant
samples from the current sample.
• A simple yet okay algorithm is LMS. Start with an
arbitrary for all i, usually =1 and =0 for all
other i.
• Let
. Then
, where
• Basically, if e< 0 while s_i>0, it means that c_i is
not large enough…
• The idea is to minimize the mean square of the
error.
Review
• A signal can also be represented in the
frequency domain.
Review
• LTI system.
– A linear system is a system such that
– Suppose
invariant” if
. A linear system is “time
.
Review
• DFT:
• IDFT:
• Convolution:
CDMA
• Code Division Multiplexing
CDMA
• Used in 3G networks.
• Direct Sequence Spread Spectrum: spread a
data bit into multiple chips.
• Each sender has a unique chip sequence, that
is *orthogonal* with other chip sequences.
Simple Examples of CMDA
•
•
•
•
A: (-1 -1 -1 +1 +1 -1 +1 +1 )
B: (-1 -1 +1 -1 +1 +1 +1 -1 )
C: (-1 +1 -1 +1 +1 +1 -1 -1 )
D: (-1 +1 -1 -1 -1 -1 +1 -1 )