Nyquist - shannon sampling theorem example

In summary, the sampling frequency for the given signal x(t) = 3sin(9πt) - 6cos(8πt) would be twice the highest frequency component, which is 9, resulting in a Nyquist rate of 18. The overall frequency is not a commonly used term in this context.
  • #1
Jncik
103
0

Homework Statement


What sampling frequency would you use to sample the following signal

x(t) = 3sin(9πt) - 6cos(8πt)

Homework Equations

The Attempt at a Solution



let T1 be the period of 3sin(9πt) and T2 the period of 6cos(8πt)

T1 = 2π/9π = 2/9
Τ2 = 2π/8π = 2/8

thus, if f1 is the frequency of 3sin(9πt) and f2 the frequency of 6cos(8πt) we have

f1 = 1/T1 = 9/2 = 4.5
f2 = 1/T2 = 8/2 = 4

now about the niquist rate,

I'm not sure whether it will be 9(2*f1, since f1>f2) or the frequency that is 2 times larger than the overall frequency of x(t)

can you please remind this to me? thanks
 
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  • #2


I'm not sure whether it will be 9(2*f1, since f1>f2) or the frequency that is 2 times larger than the overall frequency of x(t)
Not sure what you mean by the "overall frequency of x(t)".

When you sample at the Nyquist frequency, you need to sample at twice the highest frequency component of the signal.
 
  • #3


thanks lewando

What I was meaning is that

x(t) may have two components but the overall frequency is the least common multiple of the frequencies of each of these 2 components right? so the result will be different

thus if I understand correctly the result should be 2*f1 = 9 because f1 > f2 right?
 
  • #4


...the overall frequency is the least common multiple of the frequencies of each of these 2 components right?
Honestly, I have never seen the term "overall frequency", as you have defined, used anywhere.

thus if I understand correctly the result should be 2*f1 = 9 because f1 > f2 right?
Right! :smile:
 
  • #5


thanks a lot ;)
 

1. What is the Nyquist-Shannon sampling theorem?

The Nyquist-Shannon sampling theorem, also known as the Nyquist criterion or Nyquist sampling rate, is a fundamental concept in signal processing that states that in order to accurately reconstruct a continuous signal from its sampled version, the sampling rate must be at least twice the highest frequency component of the signal.

2. Can you provide an example of the Nyquist-Shannon sampling theorem?

For example, if a signal has a maximum frequency of 100 Hz, the sampling rate must be at least 200 samples per second in order to accurately reconstruct the original signal.

3. What is the importance of the Nyquist-Shannon sampling theorem in practical applications?

The Nyquist-Shannon sampling theorem is essential in many practical applications, especially in digital signal processing, telecommunications, and data compression. It ensures that the original signal can be accurately reconstructed from its sampled version, preventing loss of information and distortion.

4. Are there any limitations to the Nyquist-Shannon sampling theorem?

Yes, the Nyquist-Shannon sampling theorem assumes that the signal is band-limited, meaning that it has no frequency components above a certain cutoff frequency. In reality, most signals are not strictly band-limited, and this can lead to errors in reconstruction.

5. How is the Nyquist-Shannon sampling theorem related to the concept of aliasing?

Aliasing is a phenomenon that occurs when the sampling rate is lower than the Nyquist rate, resulting in the loss of high-frequency components and the appearance of false low-frequency components in the reconstructed signal. The Nyquist-Shannon sampling theorem helps prevent aliasing by setting a minimum sampling rate requirement.

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