Sampling and aliasing week 8

Question 1

What does FIR filters stand for?

Answer 1

Finite impulse response filter

Question 2

Determine a peak location by:

Answer 2

(𝜔𝑡+𝜑)=0

Question 3

Whats zero crossing?

Answer 3

Zero crossing is T/4

Question 4

How much is + and - peaks spaced by?

Answer 4

Positive & Negative peaks spaced by

T/2.

Question 5

How to calc the period in (s)?

Answer 5

1/f or 2*pi/omega

Question 6

How to calculate the time shift?

Rearrange to find the shift phase?

Answer 6

𝑡(m)= - 𝜑/𝜔

𝜑=−𝜔𝑡(m)

Question 7

Whats a low pass filter?

Answer 7

passes low frequencies and attenuate high

Question 8

Whats a high pass filter?

Answer 8

passes high frequencies and attenuate low

Question 9

Whats a band pas filter?

Answer 9

passes a band of frequencies and attenuate the

rest

Question 10

Whats the phase angle in an ideal filter?

Answer 10

0

Question 11

Whats sampling?

Answer 11

Sampling is the bridge between continuous-time and discrete- time signals. We often have a continuous time signal that we wish to process in discrete time using digital computers.

Question 12

Whats some examples of sampling?

Answer 12

• Video (Digital: sampled in both space & time, Analog: Sampled in time)
• Digital images
• Digital music

Question 13

Some examples of system process signals?

Answer 13

• Change x(t) into y(t)
• For example more BASS
• Improve x(t), e.g., image deblurring
• Extract information from x(t)

Question 14

System implementation examples?

Answer 14

• Analog/electronic:
• Circuits: resistors, capacitors, op-amps
• Digital/microprocessor
• Convert x(t) to numbers stored in memory

Question 15

Sampling notes:

Answer 15

• When a signal is sampled, it is inherently band-limited in frequency.
• In other words, when signal is sampled by a finite number of points, it cannot represent an infinite range of frequencies.
• A conventional D-to-A converter for audio will only create signals within a specific frequency range that is determined by the sampling rate.

Question 16

Why is it impossible to recover x(t) from X(d)[n]?

Answer 16

It would be nice to be able to perfectly recover x(t) from X(d)[n] , but this is in general impossible. There are an infinite number of x(t) that could produce the same sampled X(d)[n]

Question 17

What the sampling process?

Answer 17

• Sampling process
• Convert x(t) to numbers x[n]
• “n” is an integer; x[n] is a sequence of values
• Think of “n” as the storage address in memory

Question 18

What do we know about uniform sampling?

Answer 18

• Uniform sampling at t=nTs

* Ideal: x[n]=x(nTs)

Question 19

What do we know about Sampling rate (fs)?

Answer 19

• Sampling rate (fs)
• fs=1/Ts
• number of samples per second
• Uniform sampling at t=nTs=n/fs • Ideal: x[n]=x(nTs]=x(n/fs)

Question 20

What an we conclude about the sampling theorem?

Answer 20

It turns out that if x(t) is band-limited we can uniquely recover x(t) from its samples, provided we sample often enough! This is known the sampling theorem.

Question 21

What do we know about impulse train sampling?

Answer 21

• Represents the sampling of a continuous-time signal at regular intervals.
• Impulse-train sampling is multiplication of a periodic impulse train with the continuous-time signal x(t).
• The periodic impulse train p(t) is referred to as the sampling function, the period T as the sampling period, and the fundamental frequency of p(t), 𝛚s=2𝛑/T, as the sampling frequency.

Question 22

Whats the nyquist rate?

Answer 22

• So if 𝜔)(our sampling rate) is large enough that replicas of 𝑋(𝑗𝜔) don’t
overlap in 𝑋( jω , we can perfectly recover x(t) from its samples.
• How big does 𝛚s need to be to avoid overlap?
• Ifx(t)isbandlimitedwith𝑋 𝑗𝜔 =0for 𝜔 >𝜔,thenwerequire:
• 𝜔)>2𝜔: Nyquist rate
• In other words, we need to sample at a frequency greater than twice
the highest frequency component in x(t).

Question 23

Whats interpolation?

Answer 23

Interpolation is the process of fitting a continuous signal to a set of sample values. It is a commonly used method for reconstructing a function from its samples.
• The zero-order hold we just described can be used as a crude interpolation scheme:

Question 24

When do we get aliasing?

Answer 24

• As we have discussed, if we are undersampling or sampling at below the Nyquist rate, we get aliasing in the frequency domain. What does this mean?

Question 25

when does aliasing occur?

Answer 25

Aliasing refers to when these replicas overlap in frequency, and (again) occurs when 𝜔(s) < 2𝜔(M)

Question 26

What do we know if aliasing is present?

Answer 26

• If aliasing is present, an ideal lowpass filter can no longer reconstruct our original x(t) perfectly from the samples. Our spectrum is corrupted by the aliasing of the side lobes into the spectrum of 𝑋 𝑗𝜔 . However, we can show that our reconstructed signal Xr(t) is equal to x(t) at each sample point.
• That is: 𝑋3(𝑛𝑇) =x(nT), for all integer n.
• Consider this example to help us understand the relationship
between 𝑋3 𝑡 𝑎𝑛𝑑 x(t) when we are undersampled (𝜔0<2𝜔1).

Question 27

As 𝝎0 gets bigger, what can we notice about the aliased deltas?

Answer 27

• As 𝝎0 gets bigger, notice that these “aliased” deltas walk towards the origin.
• In other words, a high frequency 𝝎0 can masquerade as a lower frequency if we are undersampled through aliasing.

Question 28

A common example:

Answer 28

• In car commercials, it appears as though the wheels on the car are rotating backwards while the car is travelling forwards.
• The “camera trick” in the video that causes this illusion has a conceptual similarity with aliasing.
• The “snapshots” of the wheels position are not sufficiently fast to capture wheel’s true rotation.
• If the camera captures one frame while the wheel rotates 359degrees, the video will appear as though the wheel has travelled 10 in the opposite direction.

Question 29

What do we know about omega(m)?

Answer 29

omega(m) is the largest frequency in the sample

Question 30

What is a must when adding 2 sinusoids?

Answer 30

-They must have the same frequency