development of UWB radios because negating ultra-
short pulses were difficult to implement. Another mod-
ulation scheme that does not require pulse negation is
the so termed on-off keying (OOK), where symbol “1”
is represented by transmitting a pulse, and “0” by
transmitting nothing.
As pulse negation became easier to implement, pulse
amplitude modulation (PAM) attracted more attention.
In particular, when
M = 2
, antipodal pulses are used to
represent binary symbols, as in binary phase shift key-
ing (BPSK) or bipolar signaling. Biorthogonal signal-
ing by combining orthogonal PPM with binary PAM as
well as orthogonal waveform and block orthogonal
modulation schemes have also been reported [43].
To allow for multi-user access (MA) to the UWB
channel, time hopping (TH) was introduced early in
[52]. With TH, each pulse is positioned within each
frame duration
T
f
according to a user-specific TH
sequence
c
TH
u
(n)
. Specifically, dividing each frame into
N
c
chips each of duration
T
c
, the
u
th user’s TH code
c
TH
u
(n) ∈ [0,N
c
− 1]
corresponds to a time shift of
c
TH
u
(n)T
c
during the
n
th frame [72]. Consequently,
the
u
th user’s transmitted waveform is given by
v
u
(t ) =
E
u
+∞
n=0
a
u
n/N
f
· p
t − nT
f
− c
TH
u
(n)T
c
− b
u
n/N
f
,
(2)
where
E
u
is the
u
th user’s energy per pulse at the trans-
mitter end. With
s
u
(k) ∈ [0, M − 1]
denoting the
M
-
ary information symbol transmitted by the
u
th user
during the
k
th symbol duration, (2) subsumes several
modulation schemes. When
b
u
(k) = s
u
(k)
, and
a
u
(k) = 1
, (2) describes UWB transmissions with
M
-
ary PPM; when
a
u
(k) = 2s
u
(k) + 1 − M
, and
b
u
(k) = 0
, (2) models
M
-ary PAM. With binary sym-
bols, and
b
u
(k) = 0
,
a
u
(k) = 2s
u
(k) − 1
corresponds to
BPSK, and
a
u
(k) = s
u
(k)
corresponds to OOK [32],
[43], [69].
With TH codes, MA is achieved by altering the
pulse position from frame to frame, according to the
sequence
c
TH
u
(n)
. MA can also be enabled by modify-
ing the pulse amplitude from frame to frame. The latter
allows for many other choices of alternative spreading
codes which, individually or in combination with TH
codes, give rise to the following transmitted waveform
[c.f.
(2)
]:
v
u
(t ) =
E
u
+∞
n=0
a
u
n/N
f
c
u
(n)
· p
t − nT
f
− c
TH
u
(n)T
c
− b
u
n/N
f
,
(3)
where
c
u
(n)
is the user-specific “amplitude code” dur-
ing the
n
th frame. Depending on the spreading codes
employed, the UWB systems are termed TH-UWB
[52], direct-sequence (DS)-UWB [15], or baseband
single-carrier/multicarrier (SC/MC)-UWB [66], [77],
just to name a few.
In addition to facilitating multiple access, spreading
codes also shape the transmit spectrum. Analytical
expressions and simulated power spectral density
(PSD) for UWB transmissions are pursued in [44]
and [70]. A PSD expression for TH-UWB allowing
for both short and long spreading codes is derived
in [44]. The effects of timing jitter on PSD of UWB
transmissions utilizing random TH codes and/or
M
-ary modulation can be found in [70]. In particu-
lar, let us consider “short” spreading codes that are
symbol-periodic with period
N
f
. Let us now define
the symbol level transmitted waveform for user
u
as
IEEE SIGNAL PROCESSING MAGAZINE
NOVEMBER 2004 31
▲ 4. (a) Generally adopted pulse shapes in UWB communications; (b) Fourier transform of several pulse shapes. Pulse width:
0.7
ns.