Understanding Signals and Systems: Sampling, DFT, and Aliasing Explained

[LM741]

hey guys.
Just wanted to know if you guys agree with my answers:

An analogue signal of 1 s duration is sampled at 512 equally spaced times and its DFT is computed.

a) what is the separation in rad/s between the successive frequency components?
My ans: 2pi/512

b)what is the highest frequency present in the spectrum?
My ans: not to sure about this, but i would say it the inverse of the duration of the signal (1 second) so it is 1 Hz...is this righ?

c)What is the hight frequency that can be allowed in the analogu signal if aliasing is to be prevented?
My ans: Nyquist sampling theorm must be obeyed to avoid aliasing. Therefore, ws > 2wb, where ws is my sampling frequency and wb is my bandwidth (or maximum frequency component).
fs = 1/Ts =512 therefore ws = 2*pi*512.
therefore the highest allowed freqeuncy is: wb < ws/2
wb < 2*pi*256. if my highest frequecy is exactly 2*pi*256 i will still get aliasing right?

thanks

 

[berkeman]

-b- The highest frequency seems like it would be a signal that alternated every sample...

 

[LM741]

thanks...um could you elaborate on that one :)

 

[berkeman]

I just meant that the highest frequency that could be present in any digital spectrum is from a signal that alternates every cycle. So if your sampling rate is 512Hz, then the fastest signal you could get would be 256Hz, where the signal is alternating something like +/-1V or something every sample. I don't know if it's on purpose that this coincides with the Nyquist rate or not...

 

[LM741]

i see where you gettin at (i think..)- but the thing is - if you had to look at it from a digtal point of veiw and you knew the sampling frequency - there is no way you can determine the highest frequecny component...the most you can do (assuming nyquist theorem is obeyed) is deduce that the highest freq component is smaller then ws/2...

 

[doodle]

(a) Assuming that you don't zero-pad the signal, then the answer is correct.

(b) Taking 512 equally spaced samples of a 1s signal means that your sampling frequency is Fs = 1/(1/512) = 512Hz. Highest frequency present in the spectrum is Fs/2 = 256Hz (which corresponds to pi in rad/s).

(c) Aliasing will not occur if the signal contains no components of frequencies above or equal to Fs/2.

 

[LM741]

sup doodle - thanks
once you evaluate that the sampling frequency is 512Hz, why do you go and divide it by 2 to 256Hz to yield your highest frequency component?

 

[es1]

See Berkeman's #4 post.
It takes a minimum of 2 samples to see a change in the signal. So the highest frequency component is one that oscilates most often, but you can only see this every other sample, so divide by two. Anything higher frequency will exist between your samples and will be invisible to you.

 

[LM741]

thanks guys - still not entirely sure what you guys mean but i'll give it some thought and query again if i still don't get.
thanks

 

[doodle]

sup doodle - thanks
once you evaluate that the sampling frequency is 512Hz, why do you go and divide it by 2 to 256Hz to yield your highest frequency component?

Consider the sinusoidal x1 = cos(2pi*t), with period T = 1s. If you were to take samples at every 0.25s (i.e., a sampling rate of Fs = 4Hz), you'd have the set of amplitudes {1, 0, -1, 0, 1, ...}. Consider now another sinusoidal x2 with period 1/Fs = 0.25s, i.e., x2 = cos(8pi*t). At the same sampling rate (Fs), the samples obtained are {1, 1, 1, ...} which is similar to what you'd get if you were to sample the x3 = 1 (or dc) signal. Thus, signals such as x2 cannot be "properly represented" with DFT (at Fs) since they are indistinguishable from x3.

The highest frequency component which can be "properly represented" in the DFT spectrum (at Fs) is Fs/2, e.g., x4 = cos(4pi*t) which admits the samples {1, -1, 1, -1, 1, ...}.

Think about it.

 

[LM741]

thanks doodle- appreciate it!
but the question just asks for the highest frequency component present in the signal - they don't seem to ask for the highest frequency component that can "properly represent" the signal...
is it the same thing? can i just accept as a general rule that the highest frequency component is Fs/2. Where Fs does not have to be a value that will get perfect signal reconstruction - i.e. - its always Fs/2
one more thing - with Fs/2, this says that my highest frequency is dependent on my sampling frequency??
i mean if i ask what's the highest frequency component of cos(2*pi*t), the answer is 2*pi and that's it! its fixed!
...i hope you understand my confusion..
thanks again

 

[doodle]

The highest frequency that is preserved by a sampling rate of Fs is Fs/2. The highest frequency in a signal is signal-dependent ultimately- it has nothing to do with Fs.

It is true that the highest frequency preserved after sampling is dependent on the sampling rate. So if you want to "properly" sample the signal cos(2pi*t), with 1Hz or (or 2pi rad/s) being the highest frequency (and only) component of the signal, you'd need a sampling rate greater than 2Hz.

I hope that clears your confusion.

 

[LM741]

thanks a lot doodle!
one LAST question:
So if you want to "properly" sample the signal cos(2pi*t), with 1Hz or (or 2pi rad/s) being the highest frequency (and only) component of the signal, you'd need a sampling rate greater than 2Hz.
for this example you used above - what if i am given a sampling frequency of 1.5Hz (so it doesn't meet the nyquist property) - is my highest freqency component still 1.5Hz/2??
thanks again

 

[doodle]

The sampling rate of 1.5Hz is only good for signals upto 1.5/2 = 0.75Hz. For cos(2pi*t), which has the frequency component 1Hz, that would be a case of undersampling, and as a result, aliasing would arise.

Check out the URL below for a brief discussion on the topic:
http://zone.ni.com/devzone/cda/tut/p/id/3000"

 

[LM741]

ok i think I've got it - but each time we have evaluated the highest frequency component of a sampled signal, we have assumed that the nyquist sampling therem has been obeyed. right? this is the only logic i can make of the Fs/2 - hope this is right because then it makes sense.
Thanks very much for all your time.