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Spectral properties of IR-UWB signaling that

uses balanced unipolar complementary sequences

as symbols

Igor Dotlic (dotlic at gmail com)

December 28, 2011

1 Signal representation

Let us use bipolar representation for bits, i.e. k-th symbol bit is bk {1, 1}.Bit scrambling is applied, so bits can be assumed to be uncorrelated. Let usdefine a observation window w(t) of duration Tw

w(t) =

1Tw

, t [Tw/2, Tw/2],

0, elsewhere.(1)

Observed transmitted signal v(t) may be represented as

v(t) = w(t) [s(t) g(t)] , (2)

where represents convolution, g(t) is IR-UWB RF pulse waveform, while

s(t) is a sequence of chips defined as

s(t) =1

2

+k=

Nc1n=0

[1 + bkz(n)] (t nTc kTs). (3)

Here, Nc is number of chips per symbol, Tc is chip duration and Ts = NcTc is asymbol duration, while z(n) {1, 1} is a balanced sequence of length Nc.

Notice that chips in (3) are unipolar and take values of zero and one. Further-more, notice that unipolar chip sequences that represent bk = 1 and bk = 1 arecomplementary and derived from a single bipolar sequence z(n) as 1

2[1 z(n)]

and 12

[1 + z(n)] respectively.

2 Sequence autocorrelationWith getting rid of square brackets (3) can be rewritten as

s(t) =1

2

+n=

(t nTc) +1

2

+k=

bk

Nc1n=0

z(n)(t nTc kTs). (4)

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Now, let us calculate the autocorrelation of s(t)

Rs() = limN

1

2N Tc

+NTcNTc

E{s(t)s(t + )} dt. (5)

Since first and second term in (4) are uncorrelated, we may add their respectiveautocorrelations to get Rs()

Rs() =1

4Tc

+n=

( nTc) +1

4Tc

Nc1n=Nc+1

Rz(n)( nTc). (6)

Here, Rz(n) is autocorrelation ofz(n)

Rz(n) = 1Nc

Nc1

k=0

z(k)z(k + n), (7)

Rz(0) = 1, Rz(n) = Rz(n); since z(n) is balanced, following property holdsNc1

n=Nc+1Rz(n) = 1 + 2

Nc1n=1

Rz(n) = 0.

3 Observed transmitted signal power spectrum

From (2), power spectrum of v(t), Pv(f) can be calculated as

Pv(f) = Pw(f) [Ps(f)Pg(f)] , (8)

where Pw(f), Ps(f) and Pg(f) represent power spectra ofw(t), s(t) and g(t) re-spectively. Since w(t) and g(t) are deterministic signals, we may write Pw(f) =

|F{w(t)}|2

= Twsinc2(f Tw) and Pg(f) = |F{g(t)}|

2. For s(t), which is

stochastic signal, we use Wiener-Khinchin theorem and write Ps(f) = F{Rs()}.Hence, applying Fourier transform to (6) and multiplying with Pg(f) yields

Ps(f)Pg(f) =1

4T2c

+n=

Pg

n

Tc

f

n

Tc

+

Pg(f)

4Tc

1 + 2

Nc1n=1

Rz(n) cos(2fnTc)

.

(9)Since Pw(f) has integral over to equal to 1 and Tw Tc, Pw(f) ap-proximately acts like (f) in convolution with the second term in (9). Hence,substituting (9) into (8) and convolving yields

Pv(f) Tw

4T2c

+n=

Pg

n

Tc

sinc2

Tw

f

n

Tc

+

Pg(f)

4Tc

1 + 2

Nc1n=1

Rz(n) cos(2fnTc)

.

(10)

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4 Conclusions

In communication setup used hight of discrete spectral components that we tryto avoid is not a function of sequence z(n) but only of duration of observa-tion window Tw (longer the window higher the discrete spectral components)

1.Hence, these spectral components cannot be mitigated by any particular choiceof z(n). The proposed solution for this spectrum shaping problem is to mod-ify the prototype hardware before or at least in the manufacturing phase andrandomly (e.g. by using thermal noise based random generator) or pseudorandomly invert 1 chips in (3). In other words, make 1 chips bipolar.Then, (3) becomes

s(t) =1

2

+k=

Nc1n=0

gk,n[1 + bkz(n)](t nTc kTs), (11)

where gk,n {1, 1}. Another possibility is to randomly or pseudorandomlyinvert symbols, i.e.

s(t) =1

2

+k=

gk

Nc1n=0

[1 + bkz(n)](t nTc kTs), (12)

where gk {1, 1}. This technique should completely whiten the Ps(f) andsolve the problem at hand with a minor complexity increase of the transmitterand without any hardware changes of the receiver.

1In FCCs Part 15 rules regulated measurements window duration is Tw 1 ms.

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