2.1. Interference Rejection
The proposed architecture employs frequency conversion as done in [
13] to coexist with other existing narrowband systems. To achieve this technique, a proper choice for the frequency of the Local Oscillator (LO) is made. Let us investigate this issue in terms of some practical numerical values. If the LO frequency is set to the center frequency of the pulse, which for simplicity is chosen to be in the middle of the band allocated by the FCC for UWB (6.85 GHz), the system acts like a zero IF system and the bandwidth is halved, that is, the band from 3.1 GHz to 10.6 GHz is transformed into a baseband signal from DC to 3.75 GHz. However, the wireless LAN (WLAN) originally residing at around 5.5 GHz is shifted to 1.35 GHz, so it is still in band. Another option is to set the oscillator frequency to this WLAN frequency, such that this interference is shifted to around zero making possible to remove it using a simple high-pass or bandpass filter.
A possible disadvantage is that the downconverted bandwidth of the UWB signal extends up to approximately 5.1 GHz and the converted 2.4 GHz interferer falls in band. A third option is to set the LO to 5.5 GHz and filter out below
0.35 GHz and above
3.1 GHz. By this way, we remove the interferences completely with a simple bandpass filter, albeit at the expense of the loss of part of the incoming frequency band, from
GHz to 10.6 GHz and from 5.15 to 5.85 GHz, which is a bandwidth of 2.7 GHz, being only 36% or equivalently 1.9 dB.
Upper Path
First pulse (reference pulse): after mixing with a cosine of angular frequency
and normalized amplitude of one, we obtain
where,
,
,
are pulse shape (real part of morlet), envelope amplitude and carrier frequency, respectively.
Assuming ideal low-pass filter using
, the signal after filtering becomes
after delaying first pulse in the delayed path
For the second pulse, which will be used as data pulse, the same analysis holds but the input is now a pulse delayed by
in the transmitter:
The signal after low-pass filter equals
Lower Path
Following the same calculation process after delaying first pulse in the delayed path:
the second pulse after low-pass filter equals
Terms in (5) and (7) yield the correlator's outputs in the upper path and lower path, respectively. Let us represent these outputs in terms of data pulse (
) and its delayed version (
) and hence (5) or (7) integrated over sample time can be written as
can be thought as the correlator output under noise free channel.
Using Parseval's theorem, (8) can be rewritten as
where,
and
are data pulse and
-delayed reference pulse, respectively.
To implement the right most term in (9), we need the frequency components of
and
at discrete position of frequency domain. This is realized by a series of analog correlators and termed as sampler in Figure
2. Due to Fourier transform properties, the frequency resolution
achievable from a time-domain signal confined in
duration is calculated by
. Thus, the sample values of
and
at discrete position
can be obtained from the sampler.
However, (9) can be written in terms of information bit represented by
,:
Under AWGN, a frequency component of a
-delayed reference pulse at
can be expressed as
where
is a signal component, and
is a noise component.
The frequency component of a data pulse at
can likewise be expressed as
where
is a signal component and
is noise.
A discrete form for (8) is
According to (10), a signal signature resides only on real-axis. Thus a receiver to implement (13) uses the following decision rule:
Let us investigate the term
in detail.
, for
upper path, can be manipulated as
in (15) can be rewritten as
, where
which comes from the real part of signal components and
In the same manner,
, where
and
In the same way, for lower path,
in (16) can be rewritten as
, where
which comes from the real part of signal components and
In the same manner,
where
and
Thus, combining (15) and (16), we get
where
Two random variables
and
are zero mean and independent each other, so the mean and variance of
are
Thus, the resultant SNR at the correlator output is
and its BER performance is
Defining
and
. Now a message point for
is located at
in the constellation diagram while a message point for
is at
and hence a decision boundary is the line to bisect the line at right angle connecting two message points. Two message points are always symmetrical with respect to origin, and the connecting line has a slope of
. Let us rotate the coordination by
, then two message points are relocated at
and
where
. Then the decision rule becomes
where
.
Mean and Variance of
:
The ratio of SNRs of two systems is
However, (24) implies that the more
θ is departed away from
, the better the performance of our proposed receiver architecture. Theoretically, two systems are identical regarding performance when
. Suppose that as it is intuitionally, the variance of
and
are same, then
yielding a reasonable improvement (double or 3 dB) in SNR performance by new system. Similar approaches can be carried out for other modulation schemes such as QPSK and QAM.