Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Sampling of Bandpass Signals
Starting point is the traditional Nyquist sampling theorem: Any signal occupying the
band −Bneg
...
People commonly interpret this that if
the highest frequency component in a signal is fMAX, you need to take at least 2fMAX samples per
second
...
In words, sampling at or
above rate 2fMAX is clearly always sufficient but in case of bandpass signals we can also use
(usually much) lower sample rate
...
It should be kept in mind that the “accessible” (Nyquist) band for any
sample rate fS is −fS /2
...
Thus the one and only fundamental principle to remember in sampling is that the
resulting signal has a periodic spectrum and any part of that spectrum can basically be
selected/used for further processing
...

In the following, we consider two cases; starting from a real-valued bandpass signal, the
resulting sample stream is either
1) real-valued or
2) complex-valued
...
This is illustrated below
...
]
Now sampling at some rate fS results in a signal where the previous spectrum is replicated at
integer multiples of the sampling rate
...

Sampling: A message signal may originate from a digital or analog source
...
The process by which the continuous-time signal is

converted into a discrete–time signal is called Sampling
...
SAMPLING

 Theorem for low-pass signals:
Statement: “If a band –limited signal g(t) contains no frequency components for ‫׀‬f‫ > ׀‬W,
then it is completely described by instantaneous values g(kT s) uniformly spaced in time
with period T s ≤ 1/2W
...


Fig: [Sampling process]
Proof: Consider the signal g(t) is sampled by using a train of impulses s δ (t)
...
sδ(t) ------------------- 1
where sδ(t) – impulse train defined by
……………………………
...
3
The Fourier transform of an impulse train is given by
……………4
Applying F
...
1 and using convolution in frequency domain property,
Gδ(f) = G(f) * Sδ (f)

Using equation 4,

…………
...
For an ideal reconstruction filter the bandwidth B is
equal to W

The output of LPF is, gR(t) = gδ(t) * hR(t)
where hR(t) is the impulse response of the filter
...
HR(f)
For the ideal LPF

then impulse response is hR(t) = 2WTs
...


Fig: [Relation between Sampling rate, Upper cutoff frequency and Bandwidth
...
1:
Consider a signal g(t) having the Upper Cutoff frequency, fu = 100KHz and
the Lower Cutoff frequency fl = 80KHz
...

Therefore we can choose
m = 5
...
In this scheme,
the band pass signal is split into two components, one is in-phase component and other is
quadrature component
...
This form of sampling is called quadrature sampling
...
The in-phase component, gI(t) is
obtained by multiplying g(t) with cos(2πfct) and then filtering out the high frequency
components
...

The band pass signal g(t) can be expressed as,
g(t) = gI(t)
...
Accordingly each component may be sampled at the rate of 2W samples per
second

Fig: [Generation of in-phase and quadrature phase samples]

 Reconstruction:
From the sampled signals gI(nTs) and gQ(nTs), the signals gI(t) and gQ(t) are obtained
...


Fig: [Reconstruction of Band-pass signal g(t)]



---