What Is the Relation Between \( x(n) \) and \( X(f) \) Given \( f = kn \)?

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Discussion Overview

The discussion explores the relationship between discrete time signals \( x(n) \) and their corresponding frequency representations \( X(f) \) through the Discrete Fourier Transform (DFT), particularly under the assumption of a linear relation between time \( n \) and frequency \( f \) defined as \( f = kn \).

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that if there is a linear relation between discrete time and frequency, then the relationship between \( x(n) \) and \( X(f) \) can be established through the DFT.
  • One participant suggests examining the DFT of a specific signal \( x(n) = c \cdot n \) to understand the transformation.
  • Another participant questions the meaning of a "linear relation between discrete time and frequency," arguing that time and frequency are typically independent.
  • It is noted that while time and frequency are generally not related, in certain systems, a linear relation can be assumed with \( f = kn \), where \( k \) is a constant.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the relationship between time and frequency, with some asserting that a linear relationship can be assumed in specific contexts, while others maintain that such a relationship does not generally exist.

Contextual Notes

The discussion includes assumptions about the nature of signals and their transformations, as well as the implications of defining a linear relationship between time and frequency, which may not hold universally.

maysam1
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Suppose that there is a linear relation between discrete time (n) and frequency (f), then what is the relatian between x(n) and X(f) (X(f) is DFT transform of x(n))?
 
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maysam said:
Suppose that there is a linear relation between discrete time (n) and frequency (f), then what is the relatian between x(n) and X(f) (X(f) is DFT transform of x(n))?

Hi maysam! Welcome to MHB! (Smile)

I guess we'll need to find the DFT transform of $x(n)=c\cdot n$ where $c$ is some constant.
Can you calculate it?
What is the formula for a DFT?
 
I like Serena said:
Hi maysam! Welcome to MHB! (Smile)

I guess we'll need to find the DFT transform of $x(n)=c\cdot n$ where $c$ is some constant.
Can you calculate it?
What is the formula for a DFT?

Hi, x(n) can be any signal.
 

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maysam said:
Hi, x(n) can be any signal.

Hmm... can you clarify what a "linear relation between discrete time (n) and frequency (f)" means?

Presumably we have a signal amplitude $x(n) = x(t_n)$ that depends on time, which transforms to a signal amplitude $X(f) = X(f_k)$ that depends on frequency.
There is no relation between time and frequency.
 
I like Serena said:
Hmm... can you clarify what a "linear relation between discrete time (n) and frequency (f)" means?

Presumably we have a signal amplitude $x(n) = x(t_n)$ that depends on time, which transforms to a signal amplitude $X(f) = X(f_k)$ that depends on frequency.
There is no relation between time and frequency.

Yes, originally There is no relation between time and frequency but In a system we can assume that f=kn and k is a constant. this means that in this system we can see a linear relation between time and frequency.
 

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