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