# What is the difference between continuous time Fourier transform and discrete-time Fourier transform?

## What is the difference between continuous time Fourier transform and discrete-time Fourier transform?

Continuous Time Fourier Series is for signals which are periodic and continuous in time domain. It’s discrete and aperiodic in frequency domain. Discrete Time Fourier Transform is for signals which are aperiodic and discrete in time domain.

## What are the differences between discrete time Fourier transform DTFT and discrete Fourier transform DFT )? Explain DFT in detail with the help of examples and diagrams?

DTFT is an infinite continuous sequence where the time signal (x(n)) is a discrete signal. DFT is a finite non-continuous discrete sequence. DFT, too, is calculated using a discrete-time signal. DFT has no periodicity.

Why DFT is used?

The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal Processing. For example, human speech and hearing use signals with this type of encoding. Second, the DFT can find a system’s frequency response from the system’s impulse response, and vice versa.

Why is discrete-time Fourier transform periodic?

The discrete-time Fourier transform (DTFT) gives us a way of representing frequency content of discrete-time signals. Due to discrete-time nature of the original signal, the DTFT is 2π-periodic.

### What are the advantages of discrete Fourier transform over discrete time Fourier transform?

advantages of dft method DFT is completely discrete both in frequency and time, however DTFT is only discrete in time but continueous in frequency. DTFT is not used in actual digital signal processing, its DFT which is mostly calculated using fast algorithms called FFT algorithms.

### What is Fourier series and why it is used?

Fourier series is used to describe a periodic signal in terms of cosine and sine waves. In other other words, it allows us to model any arbitrary periodic signal with a combination of sines and cosines. How do you solve a Fourier series? Step 2: Estimate for n=0, n=1, etc., to get the value of coefficients.

What is the purpose of Fourier series?

The transform of a real-valued function ( fRE+fRO) is the even symmetric function FRE+i FIO.

• The transform of an imaginary-valued function ( i fIE+i fIO) is the odd symmetric function FRO+i FIE,and the converse is true.
• The transform of an even-symmetric function ( fRE+i fIO) is the real-valued function FRE+FRO,and the converse is true.
• Why does the Fourier series use cosine and sine?

Why does the Fourier series use cosine and sine? – Quora. Cosine and sine form an orthogonal basis for the space of continuous, periodic functions. The more similar it is to cosine, the less it is to sine, and vice versa (this is the orthogonality mentioned above).

#### Whose Fourier series are we Finding?

Recall that when we find the Fourier sine series of a function on 0 ≤ x ≤ L we are really finding the Fourier sine series of the odd extension of the function on − L ≤ x ≤ L and then just restricting the result down to 0 ≤ x ≤ L. For a Fourier series we are actually using the whole function on − L ≤ x ≤ L instead of its odd extension.