Sampling

Question 1

Sampling is time discretization

Answer 1

Converts a continuous time (CT) signal to a discrete time (DT) signal
▪ The result is a sequence of samples

Question 2

The sampling instants are defined by a clock signal (𝑇s=?

Answer 2

(𝑇𝑠 = 1/𝑓𝑠
)
▪ The clock signal controls an electronic switch (e.g., MOS transistor)
The sampled signal is stored as a voltage on a capacitor

Question 3

𝑇𝐷 𝑠𝑡𝑒𝑝 =

Answer 3

Δ𝑡 = 1/𝑓𝑠 = 1/𝐹𝐷 𝑝𝑒𝑟𝑖𝑜d

Question 4

𝐹𝐷 𝑠𝑡𝑒p =

Answer 4

Δ𝑓 = 1/𝑇𝑜=1/𝑇𝐷 𝑝𝑒𝑟𝑖𝑜d

Question 5

Sampling causes “images” in the frequency domain

Answer 5

The sampled signal is folded around 𝑓𝑠 and its multiples

▪ The part from 0 to 𝑓𝑠/2 is the only part that has physical meaning

Question 6

Aliasing

Answer 6

Aliasing is an effect that causes different signals to become indistinguishable (or aliases of
one another) when sampled.

Question 7

Nyquist Criterion:

Answer 7

𝑓𝑠 > 𝑓𝑁𝑦𝑞 = 2 × 𝐵𝑊
NOT
𝑓𝑠 > 2 × 𝑓𝑚𝑎x

Question 8

Subsampling

Answer 8

The sampled signal does not have to be a baseband signal
❑ Subsampling can be used to demodulate (down-convert) an RF signal
❑ A.k.a. Under-sampling, Harmonic Sampling, Bandpass Sampling, IF Sampling, Direct IF-toDigital Conversion

Question 9

Anti-Aliasing Filter (AAF)

Answer 9

Anti-aliasing filters are active or passive (CT or DT?) filters.
▪ The signal must be filtered before sampling (time discretization)

Question 10

Alias Band Suppression

Answer 10

Working at the limit of Nyquist criterion requires an ideal filter that does not exist.
❑ Signals in the alias band (𝑓𝑠 − 𝐵𝑊 𝑡𝑜 𝑓𝑠
) will alias in the desired signal band after sampling.
▪ Must be suppressed by AAF.
❑ Each pole gives a roll-off slope of 20 dB/decade = 6 dB/octave

Question 11

Oversampling

Answer 11

Oversampling relaxes requirements on baseband antialiasing filter

Question 12

Decimation

Answer 12

Decimation is the process of reducing the sample rate of a signal.
❑ Unless the signal is already filtered and oversampled, digital filtering is necessary.

Question 13

Reconstruction Filter

Answer 13

ADC needs an anti-aliasing filter.
❑ DAC needs a reconstruction (smoothing) filter.
▪ TD: The reconstruction filter “interpolates/restores/reconstructs” the signal.
▪ FD: The reconstruction filter suppresses the “images”

Question 14

Zero-Order Hold (ZOH)

Answer 14

Zero-order hold (ZOH) keeps the value of the signal at the sample moment.
❑ The Fourier transform of ZOH is a sinc function: sinc(x) = sin(x)/x
▪ Nulls of sinc(x) at the inverse of hold time (pulse width)
❑ The zero-order hold (ZOH) performs inherent reconstruction (filtering out images).

Question 15

Passband Droop

Answer 15

ZOH suppresses images but introduces amplitude distortion.
▪ The passband distortion may be compensated by inverse-sinc response in the digital or
analog domains

Question 16

Noise in RC Circuit

Answer 16

Resistors generate white thermal noise.

▪ But the BW is always limited by a cap

Question 17

Equivalent Noise Bandwidth

Answer 17

Define an equivalent noise BW 𝐵𝑁 such that the area under a brick-wall response is the
same area under the actual spectral density curve

Question 18

Sampling Noise

Answer 18

The sampling capacitor determines noise power ➔ SNR ➔ No. of ADC bits.

Question 19

Noise Folding

Answer 19

❑ Before sampling: 𝑃𝑛 = 𝑘𝑇/𝐶 = 𝑆𝑛 𝑓 × 𝐵𝑁
❑ After sampling 𝑃𝑛 is unchanged: 𝑃𝑛 = 𝑘𝑇/𝐶 = 𝑆𝑛,𝑠𝑎𝑚𝑝𝑙𝑒𝑑 𝑓 ⋅
𝑓𝑠
2
𝑆𝑛,𝑠𝑎𝑚𝑝𝑙𝑒𝑑 𝑓 =
𝑘𝑇
𝐶
×
2
𝑓𝑠
= 𝑆𝑛 𝑓 × 𝐵𝑁 ×
2
𝑓𝑠
𝑆𝑛,𝑠𝑎𝑚𝑝𝑙𝑒𝑑 𝑓 = 𝑆𝑛 𝑓 ×
2𝐵𝑁
𝑓𝑠
= 𝑆𝑛 𝑓 ×
𝜋𝐵𝑊
𝑓𝑠
❑ Noise power is unchanged, but noise density increases (noise folding).