What is the difference between continuous time and discrete time Fourier series?
The difference is pretty quickly explained: the CTFT is for continuous-time signals, i.e., for functions x(t) with a continuous variable t∈R, whereas the DTFT is for discrete-time signals, i.e., for sequences x[n] with n∈Z.
What is continuous time Fourier series?
The continuous-time Fourier series expresses a periodic signal as a lin- ear combination of harmonically related complex exponentials. Alternatively, it can be expressed in the form of a linear combination of sines and cosines or sinusoids of different phase angles.
What is continuous time Fourier transform?
Continuous time Fourier transform of x(t) is defined as X(ω)=∫−∞+∞x(t)e−jωtdt and discrete time Fourier transform of x(n) is defined as X(ω)=Σ∀nx(n)e−ωn. From: Nonlinear Digital Filters, 2007.
What is the relationship between continuous time signals and discrete-time signals?
A signal is considered to be a continuous time signal if it is defined over a continuum of the independent variable. A signal is considered to be discrete time if the independent variable only has discrete values.
What are the properties of FFT?
Fourier Transforms Properties
- Linearity Property. Ifx(t)F. T⟷X(ω) &y(t)F.
- Time Shifting Property. Ifx(t)F. T⟷X(ω)
- Frequency Shifting Property. Ifx(t)F. T⟷X(ω)
- Time Reversal Property. Ifx(t)F. T⟷X(ω)
- Differentiation and Integration Properties. Ifx(t)F. T⟷X(ω)
- Multiplication and Convolution Properties. Ifx(t)F. T⟷X(ω)
What is the importance of discrete-time Fourier series?
This discrete-time Fourier series representation provides notions of frequency content of discrete-time signals, and it is very convenient for calculations involving linear, time-invariant systems because complex exponentials are eigenfunctions of LTI systems.
What are the properties of discrete Fourier series?
Properties of Discrete-Time Fourier Transform
| Property | Discrete-Time Sequence | DTFT |
|---|---|---|
| Notation | x2(n) | X2(ω) |
| Linearity | ax1(n)+bx2(n) | aX1(ω)+bX2(ω) |
| Time Shifting | x(n−k) | e−jωkX(ω) |
| Frequency Shifting | x(n)ejω0n | X(ω−ω0) |
Why FFT is called fast?
It’s called FFT because the Fourier transform “may be computed much more rapidly than by other algorithms” according to Gentleman & Sande. So what is this question asking for? It’s called the Fast Fourier transform because its a fast method of calculating a Fourier transform.
What is CT and DT signal?
Continuous-time (CT) signals are functions from the reals, ℜ, which take on real values; and discrete-time (DT) signals are functions from the integers Z, which take on real values. Definition A continuous time signal is bounded if. there exists an M such that for all t ∈ ℜ |x(t)| ≤ M .
What is difference between continuous and discrete-time signal?
Continuous signal is the signal which has continuous value between two defined time, while discrete signal has only discrete amount of value in discrete equally-spaced time.
What is a continuous time system?
Continuous-Time Systems A continuous-time system is a system in which the signals at input and output are continuous-time signals. This chapter connects signals with systems, especially the study of linear time-invariant dynamic systems.
What is the advantage of FFT?
The fast Fourier transform (FFT) is a computationally efficient method of generating a Fourier transform. The main advantage of an FFT is speed, which it gets by decreasing the number of calculations needed to analyze a waveform.
What is the relationship between Fourier transform and Fourier series representation of a continuous function?
Key Concept: Relationship between Fourier Series and Fourier Transform. Note: The Fourier Transform of xT(t) is given by: XT(ω)=2π+∞∑n=−∞cnδ(ω−nω0) X T ( ω ) = 2 π ∑ n = − ∞ + ∞ c n δ ( ω − n ω 0 ) . These relationships are spelled out on a one-page pdf.