Sampling and aliasing week 8 Flashcards
What does FIR filters stand for?
Finite impulse response filter
Determine a peak location by:
(ππ‘+π)=0
Whats zero crossing?
Zero crossing is T/4
How much is + and - peaks spaced by?
Positive & Negative peaks spaced by
T/2.
How to calc the period in (s)?
1/f or 2*pi/omega
How to calculate the time shift?
Rearrange to find the shift phase?
π‘(m)= - π/π
π=βππ‘(m)
Whats a low pass filter?
passes low frequencies and attenuate high
Whats a high pass filter?
passes high frequencies and attenuate low
Whats a band pas filter?
passes a band of frequencies and attenuate the
rest
Whats the phase angle in an ideal filter?
0
Whats sampling?
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.
Whats some examples of sampling?
- Video (Digital: sampled in both space & time, Analog: Sampled in time)
- Digital images
- Digital music
Some examples of system process signals?
- Change x(t) into y(t)
- For example more BASS
- Improve x(t), e.g., image deblurring
- Extract information from x(t)
System implementation examples?
- Analog/electronic:
- Circuits: resistors, capacitors, op-amps
- Digital/microprocessor
- Convert x(t) to numbers stored in memory
Sampling notes:
- 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.
Why is it impossible to recover x(t) from X(d)[n]?
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]
What the sampling process?
- Sampling process
- Convert x(t) to numbers x[n]
- βnβ is an integer; x[n] is a sequence of values
- Think of βnβ as the storage address in memory
What do we know about uniform sampling?
- Uniform sampling at t=nTs
* Ideal: x[n]=x(nTs)
What do we know about Sampling rate (fs)?
- Sampling rate (fs)
- fs=1/Ts
- number of samples per second
- Uniform sampling at t=nTs=n/fs β’ Ideal: x[n]=x(nTs]=x(n/fs)
What an we conclude about the sampling theorem?
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.
What do we know about impulse train sampling?
- 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), πs=2π/T, as the sampling frequency.
Whats the nyquist rate?
β’ So if π)(our sampling rate) is large enough that replicas of π(ππ) donβt
overlap in π( jΟ , we can perfectly recover x(t) from its samples.
β’ How big does πs need to be to avoid overlap?
β’ Ifx(t)isbandlimitedwithπ ππ =0for π >π,thenwerequire:
β’ π)>2π: Nyquist rate
β’ In other words, we need to sample at a frequency greater than twice
the highest frequency component in x(t).
Whats interpolation?
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.
β’ The zero-order hold we just described can be used as a crude interpolation scheme:
When do we get aliasing?
β’ 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?
when does aliasing occur?
Aliasing refers to when these replicas overlap in frequency, and (again) occurs when π(s) < 2π(M)
What do we know if aliasing is present?
β’ 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 π ππ . However, we can show that our reconstructed signal Xr(t) is equal to x(t) at each sample point.
β’ That is: π3(ππ) =x(nT), for all integer n.
β’ Consider this example to help us understand the relationship
between π3 π‘ πππ x(t) when we are undersampled (π0<2π1).
As π0 gets bigger, what can we notice about the aliased deltas?
- As π0 gets bigger, notice that these βaliasedβ deltas walk towards the origin.
- In other words, a high frequency π0 can masquerade as a lower frequency if we are undersampled through aliasing.
A common example:
- In car commercials, it appears as though the wheels on the car are rotating backwards while the car is travelling forwards.
- The βcamera trickβ in the video that causes this illusion has a conceptual similarity with aliasing.
- The βsnapshotsβ of the wheels position are not sufficiently fast to capture wheelβs 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.
What do we know about omega(m)?
omega(m) is the largest frequency in the sample
What is a must when adding 2 sinusoids?
-They must have the same frequency
