# DTFT vs the spectrum of a sampled signal

• CoolDude420
In summary, the conversation discusses the confusion surrounding the differences between the spectrum of a sampled signal as derived through the Dirac delta function and the discrete-time Fourier transform (DTFT). It is clarified that the two equations are equivalent and serve different purposes - the first for illustration and the second for actual calculation.
CoolDude420

## Homework Statement

Hi,

So we started sampling/sampling theorem, dirac delta, DTFT in a digital signal processing module and I'm kinda confused. I understand how to derive the following formulae but these two formulae are so different to each other that I don't understand why?

We are first told that the spectrum of the sampled signal is,

I completely understand where this comes from (the whole dirac delta part/sifting property and FT property). Okay all makes sense now. Then we are introduced to the DTFT and again I understand the derivation but the result is really different,

Now, how can these two things be equal to each other? I mean in both cases, we were deriving the frequency spectrum of the sampled signal. Why are the answers so different?

## The Attempt at a Solution

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berkeman
You understand what ## e^{jθ} ## represents? It is ## cosθ + j sinθ ##. The fn coefficients are such that the complex terms cancel out so that you have a real signal.

I don't mean to be rude, I'm just trying to figure what you know.

scottdave said:
You understand what ## e^{jθ} ## represents? It is ## cosθ + j sinθ ##. The fn coefficients are such that the complex terms cancel out so that you have a real signal.

I don't mean to be rude, I'm just trying to figure what you know.

So essentially these two are equivalent? Would I be right in saying that the first equation is sort of good for illustration purposes and the second one helps in leading to the DFT for actually calculating?

## 1. What is the difference between DTFT and the spectrum of a sampled signal?

The Discrete-Time Fourier Transform (DTFT) is a mathematical tool used to analyze signals that are discrete in both time and frequency domains. It is a continuous function of frequency and is used to represent the frequency content of a continuous-time signal. On the other hand, the spectrum of a sampled signal is the frequency representation of a discrete-time signal obtained by sampling a continuous-time signal. It is a discrete function of frequency and is obtained by taking the Discrete Fourier Transform (DFT) of the sampled signal.

## 2. How are DTFT and the spectrum of a sampled signal related?

The DTFT and the spectrum of a sampled signal are closely related. The spectrum of a sampled signal can be obtained by sampling the DTFT of the original continuous-time signal. This means that the DTFT acts as a bridge between the continuous-time and discrete-time domains, allowing us to analyze the frequency content of a sampled signal.

## 3. What is the main advantage of using DTFT over the spectrum of a sampled signal?

One of the main advantages of using DTFT is that it allows us to analyze signals that are not necessarily periodic or have finite duration. This means that we can use the DTFT to analyze signals that are not suitable for sampling. On the other hand, the spectrum of a sampled signal can only be obtained for signals that are periodic and have finite duration.

## 4. Can the DTFT and the spectrum of a sampled signal be used interchangeably?

No, the DTFT and the spectrum of a sampled signal cannot be used interchangeably. While they are related, they represent different properties of a signal. The DTFT represents the frequency content of a continuous-time signal, while the spectrum of a sampled signal represents the frequency content of a discrete-time signal obtained by sampling the original signal. Therefore, they cannot be used interchangeably.

## 5. Are there any limitations to using the DTFT and the spectrum of a sampled signal?

Yes, there are limitations to using the DTFT and the spectrum of a sampled signal. The DTFT assumes that the signal being analyzed is finite and has a finite number of frequency components. If the signal is infinite or has an infinite number of frequency components, the DTFT cannot be used. Similarly, the spectrum of a sampled signal can only be obtained for signals that are periodic and have finite duration. Additionally, the sampling process may introduce errors and distortions in the frequency domain, limiting the accuracy of the spectrum of a sampled signal.

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