Digital Signal Processing
Moslem Amiri, Václav Přenosil
Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Periodic Sampling
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Periodic Sampling
 Periodic sampling
 Process of representing a continuous signal with
a sequence of discrete data values
 In practice, sampling is performed by applying a
continuous signal to an analog-to-digital (A/D)
converter
 Primary concern is how fast a given continuous
signal must be sampled to preserve its
information content
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Aliasing
 Example: given following sequence of values
 They represent values of a time-domain
sinewave taken at periodic intervals
 Draw that sinewave
0=x(6)0.866,-=x(5)0.866,-=x(4)0,=x(3)0.866,=x(2)0.866,=x(1)0,=x(0)
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Aliasing
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Aliasing
 Frequency ambiguity
 If data sequence represents periodic samples of
a sinewave, we cannot unambiguously determine
frequency of sinewave from those sample values
alone
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Aliasing
 Mathematical origin of frequency ambiguity
 When sampling at a rate of fs samples/second,
if k is any positive or negative integer, we
cannot distinguish between sampled values
of a sinewave of fo Hz and a sinewave of
(fo+kfs) Hz
))(2sin()2sin()(
))(2sin()(
))(2sin()22sin()2sin()(
)2sin()(
/1
if
ssoso
tf
s
s
o
knm
s
s
ososo
o
ntkffntfnx
nt
t
k
fnx
nt
nt
m
fmntfntfnx
tftx
ss
+== →
+= →
+=+==
=
=
=
ππ
π
ππππ
π
7
Aliasing
 Frequency ambiguity (aliasing) effects
 Spectrum of any discrete series of sampled
values contains periodic replications of original
continuous spectrum
 Period between these replicated spectra in
frequency domain is always fs
 Spectral replications repeat all the way in both
directions of frequency spectrum
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Aliasing
9
Aliasing
 Fig. 2-2(a)
 fo = 7 kHz, fs = 6 kHz
 k = −1  fo+kfs = [7+(−1·6)] = 1 kHz
 No processing scheme can determine if
sequence of sampled values came from a 7 kHz
or a 1 kHz sinusoid
 1 kHz is an alias of 7 kHz
 Fig. 2-2(b)
 fo = 4 kHz, fs = 6 kHz
 k = −1  fo+kfs = [4+(−1·6)] = −2 kHz
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Aliasing
 Fig. 2-2(c)
 fs/2 is an important quantity, referred to by critical
Nyquist, half Nyquist, or folding frequency
 We’re interested in signal components that are
aliased into frequency band between −fs/2 and
+fs/2
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Aliasing
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Sampling Lowpass Signals
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Sampling Lowpass Signals
 Fig. 2-4(a)
 Spectrum of a continuous real-valued lowpass
x(t) signal
 Spectrum is symmetrical around zero Hz
 Signal is band-limited
 Its spectral amplitude is zero above +B Hz and below
−B Hz
 x(t) time signal is called a lowpass signal
because its spectral energy is low in frequency
 Spectrum of a continuous signal cannot be
represented in a digital machine in its current
band-limited form  replicated form of (b)
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Sampling Lowpass Signals
 Nyquist criterion
 fs ≥ 2B, to separate spectral replications at folding
frequencies of ±fs/2
 Fig. 2-4(c)
 Sampling frequency is lowered to fs = 1.5B Hz
 Lower edge and upper edge of spectral
replications centered at +fs and −fs now lie in
band of interest
 Equivalent to original spectrum folding to left at +fs/2
and folding to right at −fs/2
 Spectral information in bands of −B to −B/2 and B/2 to
B Hz is corrupted (aliasing errors)
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Sampling Lowpass Signals
 A key property of band ±fs/2 Hz
 Entire spectral content (any signal energy located
above +B Hz and below −B Hz) of original
continuous spectrum always ends up in band of
interest between −fs/2 and +fs/2 after sampling,
regardless of sample rate
 For this reason, continuous (analog) lowpass
filters are necessary in practice
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Sampling Lowpass Signals
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Sampling Lowpass Signals
18
Sampling Bandpass Signals
 Bandpass sampling
 A technique to sample a continuous bandpass
signal that is centered about some frequency
other than zero Hz
 Reduces speed requirement of A/D converters
below that necessary with traditional lowpass
sampling
 Reduces amount of digital memory necessary to
capture a given time interval of a continuous
signal
 We’re more concerned with a signal’s bandwidth
than its highest-frequency component
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Sampling Bandpass Signals
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Sampling Bandpass Signals
 Fig. 2-7(a)
 Negative frequency portion of signal is mirror
image of positive frequency portion (real signal)
 Highest-frequency = 22.5 MHz
 Nyquist criterion  sampling frequency must be
a minimum of 45 MHz
 Fig. 2-7(b)
 If sample rate is 17.5 MHz, spectral replications
are located exactly at baseband
 Sampling at 45 MHz was unnecessary to avoid
aliasing—instead we’ve used spectral replicating
effects to our advantage
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Sampling Bandpass Signals
 Sampling translation
 Bandpass sampling performs digitization and
frequency translation in a single process
 We can sample at some still lower rate and
avoid aliasing
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Sampling Bandpass Signals
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Sampling Bandpass Signals
 Fig. 2-8(a)
 Continuous input bandpass signal of bandwidth B
 Carrier frequency (signal is centered at) = fc Hz
 Sample rate = fs′ Hz  spectral replications of
positive and negative bands, Q and P, butt up
against each other at zero Hz
 m = an arbitrary number of replications in the
range of 2fc − B
 m can be any positive integer so long as fs′ is never
less than 2B
m
Bf
fBfmf c
scs
−
=−=
2
or2 ''
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Sampling Bandpass Signals
 Fig. 2-8
 If fs′ is increased, original spectra (bold) do not
shift, but all replications will shift
 At zero Hz, P band shifts to right, and Q band
shifts to left
 These replications will overlap and aliasing
occurs
 Thus, for an arbitrary m, there is a frequency that
sample rate must not exceed
m
Bf
f c
s
−
≤
2
'
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Sampling Bandpass Signals
 Fig. 2-8(b) and (c)
 If we reduce sample rate below fs′ shown in (a),
spacing between replications will decrease in
direction of arrows in (b)
 Original spectra do not shift
 At some sample rate fs″ (fs″ < fs′), replication P′ will
butt up against positive original spectrum at fc as
shown in (c)
 fs″ decreased  aliasing occurs
1
2
or2)1( ''''
+
+
=+=+
m
Bf
fBffm c
scs
1
2
''
+
+
≥
m
Bf
f c
s
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Sampling Bandpass Signals
 To avoid aliasing, fs may be chosen
anywhere in the range
(1)
 m is an arbitrary, positive integer ensuring fs ≥ 2B
1
22
+
+
≥≥
−
m
Bf
f
m
Bf c
s
c
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Sampling Bandpass Signals
 Example (Fig. 2-7(a))
 fc = 20 MHz, B = 5 MHz
 Sample rates below 11.25 MHz unacceptable
 Will not satisfy Eq. (1) as well as fs ≥ 2B
 Optimum sampling frequency is the frequency
where spectral replications butt up against each
other at zero Hz
m (2fc-B) / m (2fc+B) / (m+1) Optimum sampling rate
1 35.0 MHz 22.5 MHz 22.5 MHz
2 17.5 MHz 15.0 MHz 17.5 MHz
3 11.66 MHz 11.25 MHz 11.25 MHz
4 8.75 MHz 9.0 MHz ---
5 7.0 MHz 7.5 MHz ---
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Sampling Bandpass Signals
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Practical Aspects of Bandpass Sampling
 Spectral Inversion in Bandpass Sampling
 Some of permissible fs values from Eq. (1)
provide a sampled baseband spectrum (located
near zero Hz) that is inverted from original analog
signal’s positive and negative spectral shapes
 Happens when m, in Eq. (1), is an odd integer
 We can invert spectrum back to its original orientation
 Discrete spectrum of any digital signal can be inverted
by multiplying signal’s discrete-time samples by (−1)n
 Center of flipping is fs/4 Hz (and −fs/4 Hz)
 When original positive spectral bandpass
components are symmetrical about fc frequency,
spectral inversion presents no problem
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Practical Aspects of Bandpass Sampling
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Practical Aspects of Bandpass Sampling
 Positioning sampled spectra at fs/4
 In many signal processing applications it is useful
to use an fs bandpass sampling rate that forces
sampled spectra to be centered exactly at ±fs/4
 To ensure that sampled spectra reside at ±fs/4,
select fs using
,...3,2,1where,
12
4
=
−
= k
k
f
f c
s
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Practical Aspects of Bandpass Sampling
 Noise in bandpass-sampled signals
 Signal-to-noise ratio (SNR) is ratio of power of a
signal over total background noise power
 Negative aspect of bandpass sampling
 SNR of digitized signal is degraded
 All of background spectral noise (Fig. 2-11(b)) resides
in range of −fs/2 to fs/2 (Fig. 2-11(c))
 Bandpass-sampled background noise power increases
by a factor of m + 1 (denominator of right-side ratio in
Eq. (1)) while signal power P remains unchanged
 Bandpass-sampled signal’s SNR is reduced by
below SNR of original analog signal
dBmDSNR )1(log10 10 +⋅=
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Practical Aspects of Bandpass Sampling