What is Dtft: Definition and 18 Discussions

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis.
Both transforms are invertible. The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.

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  1. C

    DFT vs. DTFT: Understanding the Difference

    Homework Statement I'm kind of confused between DFT and DTFT. Here is my understanding: Okay, so let's say we have time domain, continuous, analogue signal from a sensor - ##x(t) ## 1. We sample this signal, giving us something like the following with an impulse train Now this is a...
  2. C

    DTFT vs the spectrum of a sampled signal

    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...
  3. L

    Understanding the discrete time Fourier transformation

    Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain. When performing a Fourier transform on a time discrete signal, we have to apply the...
  4. M

    Understanding DTFT Angular Frequency: Mike's Questions

    Hello everyone. Iam trying to understand the discrete time Fourier transform for a signal processing course but Iam quite confused about the angular frequency.If I have a difference equation given, what values should I choose for my angular frequency if I do not know anything about the sample...
  5. nacreous

    Can you calculate the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n]?

    Homework Statement There is a signal y[n] with a differentiable DTFT Y(eiw). Find the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n] (where of course i = √-1). Homework Equations Sifting property ∫eiwndw = 2π*δ[n] from [-π,π] (integral a) leads to ∫Y(eiwn)dw = 2π*y[n] from [-π,π] (integral b)...
  6. N

    Understanding the Variables of the FT and DTFT: Intuition and Differences

    Could someone explain the intuition behind the variables of the FT and DTFT? Do I understand it correctly ? For FT being X(f), I understand that f is a possible argument the frequency, as in number of cycles per second. FT can be alternatively parameterized by \omega = 2 \pi f which...
  7. D

    Inverse Discrete Time Fourier Transform (DTFT) Question

    1. Given: The DTFT over the interval |ω|≤\pi, X\left ( e^{jω}\right )= cos\left ( \frac{ω}{2}\right ) Find: x(n) 2. Necessary Equations: IDTFT synthesis equation: x(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}X\left ( e^{jω} \right ) e^{j\omega n}d\omega Euler's Identity...
  8. P

    DTFT question regarding a pair

    Homework Statement My book writes the following: using pair for the Discrete Time Fourier Transform: -a^{k}u[-k-1] <---(DTFT)---> \frac{1}{1-ae^{-iw}} for \left | a \right | > 1 Homework Equations Well, for the simple similar pair such as: a^{k}u[k] <---(DTFT)--->...
  9. Z

    Calculating the DTFT of 1: Challenges and Solutions

    Hello. I'm stuck on calculating DTFT of 1. DTFT formula is: so dtft of 1 is: 1) in case of w=1, sum becomes: doesn't it diverge? 2) in case of w no 1, how the hell should that sum be calculated? thanks
  10. C

    Types of Convergence of the DTFT & Relation to Summability of x[n]

    Given a discrete time signal x[n] that has a DTFT (which exists in the mean square convergence or in the uniform convergence sense), how can we tell if the signal x[n] converges absolutely? I know the following: x[n] is absolutely summable <=> X(e^{j \omega}) converges uniformly (i.e...
  11. J

    Can the DTFT of a Unit Step Function be Simplified for u(n) - u(n-L)?

    So I'm trying to find the DTFT of the following; where u(n) is the unit step function. u \left( n \right) =\cases{0&$n<0$\cr 1&$0\leq n$\cr} I want to find the DTFT of u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right) Which ends up being a piecewise defined function...
  12. X

    Plotting DTFT of Finite Sum in MATLAB

    Homework Statement x[n] = Ʃ ck * δ(n-k), from k = -N to N. Plot the DTFT as a function of the number of terms N. This is a finite sum. Homework Equations The equation for the DTFT of a signal, which is Ʃ x[n] * e-j*2∏*∅*n, from n = -∞ to +∞ The Attempt at a Solution I have...
  13. I

    Discrete Time Fourier Transformation (DTFT) Question

    Hello all ! Homework Statement I have the following problem. I have to calculate the DTFT of this : x(n)=u(n)-u(n-4). Homework Equations Fourier Transformations The Attempt at a Solution So far , from what I have studied I have understood, that a DTFT , is actually many...
  14. Z

    Z-Transform & DTFT: Finite-Length Signal & Two Poles

    1. Homework Statement [/b] You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|. a) x is a finite-length signal b) \hat{X} has...
  15. Z

    Z-Transform and DTFT Homework: Poles, Zeros, and Endpoints

    Homework Statement You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|. a) x is a finite-length signal b) \hat{X} has exactly two...
  16. J

    Understanding the Use of Variable Notation in the DTFT

    Can someone explain to me why sometimes I see the DTFT as functions of capital omegas or e^(jomega). I'm failing to see the reason.
  17. A

    Solving for the DTFT of (0.8)^n u[n]

    Homework Statement Compute the DTFT of the following signal. x[n] = (0.8)^n u[n] Homework Equations Properties of DTFT The Attempt at a Solution Well, my professor tells me to use the properties of DTFT to solve this. I'd love to - except I don't know what the DTFT of (0.8)^n...
  18. L

    MATLAB Finding DFT and DTFT of 12-Point Sequence with MatLab

    I'm considering the 12 –point sequence x[n] which is defined as x[n] = {1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1}. I'd like to use Matlab to find the DFT (X[k] of x[n]) and DTFT (X(e^jw) of x[n]). I realize that the DFT is sampled version of DTFT, and I want to show this graphically using...