lecture 4 sampling overview of sampling theory. sampling continuous signals sample period is t,...
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Lecture 4 Sampling
Overview of Sampling Theory
Sampling Continuous Signals
Sample Period is T, Frequency is 1/T x[n] = xa(n) = x(t)|t=nT
Samples of x(t) from an infinite discrete sequence
Continuous-time Sampling Delta function (t)
Zero everywhere except t=0 Integral of (t) over any interval including
t=0 is 1 (Not a function – but the limit of
functions) Sifting
)()()( 00 tfdttttf
Continuous-time Sampling Defining the sequence by multiple sifts:
Equivalently:
Note: xa(t) is not defined at t=nT and is zero for other t
n
a nTttxtx )()()(
n
a nTtnTxtx )()()(
Reconstruction Given a train of samples – how to
rebuild a continuous-time signal? In general, Convolve some impluse
function with the samples:
Imp(t) can be any function with unit integral…
n
nTtimpnTxtx )()()(
Example
Linear interpolation:
Integral (0,2) of imp(t) = 1Imp(t) = 0 at t=0,2Reconstucted function is piecewise-
linear interpolation of sample values
else
tttimp
0
2011)(
DAC Output Stair-step output
DAC needs filtering to reduce excess high frequency information
else
ttimp
0
101)(
Sinc(x) – ‘Perfect Reconstruction’
Is there an impulse function which needs no filtering?
Why? – Remember that Sin(t)/t is Fourier Transform of a unit impulse
TtTt
timp
)sin(
)(
Perfect Reconstruction II Note – Sinc(t) is non-zero for all t
Implies that all samples (including negative time) are needed
Note that x(t) is defined for all t since Sinc(0)=1
n
TnTtTnTt
nxtx)(
)(sin
][)(
Operations on sequences Addition: Scaling: Modulation:
Windowing is a type of modulation Time-Shift: Up-sampling: Down-sampling:
][][][ nwnxny
][][ nxAny ][][][ nwnxny
][][ 1 nxny
xd[n]x[nM ]
else,0
,2,,0],/[][
LLnLnxnxu
Up-sampling
0 10 20 30 40 50-1
-0.5
0
0.5
1Input Sequence
Time index n
Am
plitu
de
0 10 20 30 40 50-1
-0.5
0
0.5
1Output sequence up-sampled by 3
Time index nA
mpl
itude
]3/[][ nxnxu
][nxu][nx
Down-sampling (Decimation)
0 10 20 30 40 50-1
-0.5
0
0.5
1Input Sequence
Time index n
Am
plitu
de
0 10 20 30 40 50-1
-0.5
0
0.5
1Output sequence down-sampled by 3
Am
plitu
de
Time index n
]3[][ nxnxd
][nxd][nx
Resampling (Integer Case) Suppose we have x[n] sampled at
T1 but want xR[n] sampled at T2=L T1
n
nTtimpulsenTxtx )()()( 11
n
n
kTtnR
TnLkimpulsenLTx
nTkTimpulsenTx
nTtimpulsenTxkx
))(()(
)()(
)()(][
22
121
11
2
n
R nkimpulsenxkx ][][][
Sampling Theorem Perfect Reconstruction of a
continuous-time signal with Bandlimit f requires samples no longer than 1/2f Bandlimit is not Bandwidth – but limit
of maximum frequency Any signal beyond f aliases the
samples
Aliasing (Sinusoids)
Alaising For Sinusoid signals (natural
bandlimit): For Cos(n), =2k+0
Samples for all k are the same! Unambiguous if 0<< Thus One-half cycle per sample
So if sampling at T, frequencies of f=+1/2T will map to frequency