This preview shows page 1. Sign up to view the full content.
Unformatted text preview: m
otherwise
8 Then we only get the term we get is the term we want,
˜
Dn,k = Dn,k
and we ignore all of the other channels!
We want to choose φn(t) to be orthogonal signals.
There are many possible choices. For example
φn(t) = cos(nω0t)
where ω0 = 2π . The Dn,k are then the coeﬃcients of the cosine Fourier
T
series of the signal. This eﬀectively shifts each channel to a diﬀerent
frequency!
However, what we really want is something very easy to compute ... EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 9 Hadamard Waveforms
Hadamard waveforms are an orthogonal set made up of only ±1’s.
The ﬁrst couple of Hadamard signals are (in sequency order):
!0(t )
!1(t )
!2(t )
!3(t )
!4(t )
!5(t ) 1
−1
1 Tt −1
1 T −1
1 T t −1
1 T t −1
1 T t −1 T t EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly t 10 The signs of the Hadamard signals can be generated (in a diﬀerent order)
by deﬁning
11
H2 =
1 −1
and then deﬁning 1
1
1
1
1 −1
H2
H2
1 −1 H4 =
=
1
H2 −H2
1 −1 −1 1 −1 −1
1
and similarly for H8, H16, ... It is easy to show that the columns of HN are orthogonal, and hence that
the Hadamard functions with these signs are orthogonal (try this!).
We can tag each channel by assigning it a Hadamard waveform φn(t).
We can continue to add users by adding new codes (at some point we run
out of bandwidth, though).
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 11 Synchronous Detection
Hadamard decoding fails if it is not synchronized y(t )
T
y(t − )
2 −!1 !1 −!1 −!1 !1 !1 −!1 !1 −!2 !2 −!1 −!2 !1 −!1 !2 −!2 !2 !1 =
!2 = Without a delay, the signal decodes to channel 1.
With a delay, the decoding is completely diﬀerent. EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 12 The Hadamard codes only work if all of the signals are synchronized.
Fine when the basestation is talking to the handsets (the downlink). The
basestation can synchronize all the signals before transmitting. Downlink Base Station Handsets EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 13 Does not work well when the handsets are talking to the basestation (the
uplink).
Each channel has it’s own delay: Uplink Base Station Handsets Even if we could synchronize to one channel, we would be decoding the
other channels with delays, and these interfere with our channel. EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 14 Uplink Codes
Ideally, we would like a code that
• Is orthogonal, so that we can isolate a particular channel
• Has a sharp autocorrelation, so that we can accurately identify the delay
of a channel, and decode it properly.
Unfortunately, Hadamard codes have broad autocorrelations:
= = Not good for estimating delays!
EE102A:Signal Processing and Linear Systems I; Win 0910, Pauly 15 We would like autocorrelations like the Barker codes from last week, = However, there are no orthogonal sets of signals wit...
View
Full
Document
 Fall '13
 Mukamel
 Signal Processing

Click to edit the document details