Description Undersampling

spectrum of fm radio band (88–108 mhz) , baseband alias under 44 mhz (n = 5) sampling. anti-alias filter quite tight fm radio band required, , there s not room stations @ nearby expansion channels such 87.9 without aliasing.



spectrum of fm radio band (88–108 mhz) , baseband alias under 56 mhz (n = 4) sampling, showing plenty of room bandpass anti-aliasing filter transition bands. baseband image frequency-reversed in case (even n).



example: consider fm radio illustrate idea of undersampling.


in us, fm radio operates on frequency band fl = 88 mhz fh = 108 mhz. bandwidth given by








w
=

f

h


−

f

l


=
108
 

m
h
z

−
88
 

m
h
z

=
20
 

m
h
z



{\displaystyle w=f_{h}-f_{l}=108\ \mathrm {mhz} -88\ \mathrm {mhz} =20\ \mathrm {mhz} }






the sampling conditions satisfied for





1
≤
n
≤
⌊
5.4
⌋
=

⌊



108
 

m
h
z



20
 

m
h
z




⌋



{\displaystyle 1\leq n\leq \lfloor 5.4\rfloor =\left\lfloor {108\ \mathrm {mhz} \over 20\ \mathrm {mhz} }\right\rfloor }






therefore, n can 1, 2, 3, 4, or 5.


the value n = 5 gives lowest sampling frequencies interval



43.2
 

m
h
z

<

f


s



<
44
 

m
h
z



{\displaystyle 43.2\ \mathrm {mhz} <f_{\mathrm {s} }<44\ \mathrm {mhz} }

, scenario of undersampling. in case, signal spectrum fits between 2 , 2.5 times sampling rate (higher 86.4–88 mhz lower 108–110 mhz).


a lower value of n lead useful sampling rate. example, using n = 4, fm band spectrum fits between 1.5 , 2.0 times sampling rate, sampling rate near 56 mhz (multiples of nyquist frequency being 28, 56, 84, 112, etc.). see illustrations @ right.


when undersampling real-world signal, sampling circuit must fast enough capture highest signal frequency of interest. theoretically, each sample should taken during infinitesimally short interval, not practically feasible. instead, sampling of signal should made in short enough interval can represent instantaneous value of signal highest frequency. means in fm radio example above, sampling circuit must able capture signal frequency of 108 mhz, not 43.2 mhz. thus, sampling frequency may little bit greater 43.2 mhz, input bandwidth of system must @ least 108 mhz. similarly, accuracy of sampling timing, or aperture uncertainty of sampler, analog-to-digital converter, must appropriate frequencies being sampled 108mhz, not lower sample rate.


if sampling theorem interpreted requiring twice highest frequency, required sampling rate assumed greater nyquist rate 216 mhz. while satisfy last condition on sampling rate, grossly oversampled.


note if band sampled n > 1, band-pass filter required anti-aliasing filter, instead of lowpass filter.

as have seen, normal baseband condition reversible sampling x(f) = 0 outside interval:  





(
−


1
2



f


s



,


1
2



f


s



)

,



{\displaystyle \scriptstyle \left(-{\frac {1}{2}}f_{\mathrm {s} },{\frac {1}{2}}f_{\mathrm {s} }\right),}

and reconstructive interpolation function, or lowpass filter impulse response,  




sinc
⁡

(
t

/

t
)

.



{\displaystyle \scriptstyle \operatorname {sinc} \left(t/t\right).}

to accommodate undersampling, bandpass condition x(f) = 0 outside union of open positive , negative frequency bands

(
−


n
2



f


s



,
−



n
−
1

2



f


s



)

∪

(



n
−
1

2



f


s



,


n
2



f


s



)



{\displaystyle \left(-{\frac {n}{2}}f_{\mathrm {s} },-{\frac {n-1}{2}}f_{\mathrm {s} }\right)\cup \left({\frac {n-1}{2}}f_{\mathrm {s} },{\frac {n}{2}}f_{\mathrm {s} }\right)}

positive integer



n



{\displaystyle n\,}

.
which includes normal baseband condition case n = 1 (except intervals come @ 0 frequency, can closed).

the corresponding interpolation function bandpass filter given difference of lowpass impulse responses:

n
sinc
⁡

(



n
t

t


)

−
(
n
−
1
)
sinc
⁡

(



(
n
−
1
)
t

t


)



{\displaystyle n\operatorname {sinc} \left({\frac {nt}{t}}\right)-(n-1)\operatorname {sinc} \left({\frac {(n-1)t}{t}}\right)}

.

on other hand, reconstruction not goal sampled if or rf signals. rather, sample sequence can treated ordinary samples of signal frequency-shifted near baseband, , digital demodulation can proceed on basis, recognizing spectrum mirroring when n even.

further generalizations of undersampling case of signals multiple bands possible, , signals on multidimensional domains (space or space-time) , have been worked out in detail igor kluvánek.