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Q: What is short time fourier transform and what are its properties?

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the main application of fourier transform is the changing a function from frequency domain to time domain, laplaxe transform is the general form of fourier transform .

A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.

The Fourier transform is a mathematical transformation used to transform signals between time or spatial domain and frequency domain. It is reversible. It refers to both the transform operation and to the function it produces.

Spectral analysis of a repetitive waveform into a harmonic series can be done by Fourier analyis. This idea is generalised in the Fourier transform which converts any function of time expressed as a into a transform function of frequency. The time function is generally real while the transform function, also known as a the spectrum, is generally complex. A function and its Fourier transform are known as a Fourier transform pair, and the original function is the inverse transform of the spectrum.

A fast fourier transform is an algorithm that converts time or space to frequency, or vice versa. They are mainly used in engineering, math and sciences.

fourier transform deals with continuous time non-periodic signals(if the input is continuous time periodic signal(time domain) then the output is continuous nonpeiodic signal (frequency domain)) DTFT-input-discrete time nonperoidic signal(time domain),output-continuous peroidic signal(frequency domain)

z transform is related to discrete time signal while fourier series is related to continuous time signal. z transform=sigmalm -infinty to +infinity x(n)z-n

A Fourier series is a series of sine and cosine harmonics of a particular frequency. For example sinf+icosf + 3 sin2f+ 5icos2f... where the successive terms are multiples of the fundamental frequency f. It is typical ( but as far as I know not required) that complex numbers are used. A Fourier transform converts a time domain wave form (like a sound wave) into the coefficients of the corresponding Fourier series. A DFT is a digital approximation to a Fourier transform, usually using something like the Cooley-Tuckey Fast Fourier Transform (FFT) for efficiency. The underlying Fourier theorem is something like: Every bounded periodic continuous (needed to avoid Gibbs) function , or wave form, can be written as the sum of its Fourier series. i.e. It is a sum of sines and cosines In otherwords, you take a wave form in the time domain like a sound wave and break it into its components (various frequencies) by the Fourier Transform. The results of the Transform are the coefficients of the Fourier series. The wave form of a voice converted to components (and perhaps a little more) is a voiceprint.

The Fourier series is an expression of a pattern (such as an electrical waveform or signal) in terms of a group of sine or cosine waves of different frequencies and amplitude. This is the frequency domain. The Fourier transform is the process or function used to convert from time domain (example: voltage samples over time, as you see on an oscilloscope) to the frequency domain, which you see on a graphic equalizer or spectrum analyzer. The inverse Fourier transform converts the frequency domain results back to time domain. The use of transforms is not limited to voltages.

it is used for linear time invariant systems

The fourier transform is used in analog signal processing in order to convert from time domain to frequency domain and back. By doing this, it is easier to implement filters, shifters, compression, etc.

Laplace = analogue signal Fourier = digital signal Notes on comparisons between Fourier and Laplace transforms: The Laplace transform of a function is just like the Fourier transform of the same function, except for two things. The term in the exponential of a Laplace transform is a complex number instead of just an imaginary number and the lower limit of integration doesn't need to start at -∞. The exponential factor has the effect of forcing the signals to converge. That is why the Laplace transform can be applied to a broader class of signals than the Fourier transform, including exponentially growing signals. In a Fourier transform, both the signal in time domain and its spectrum in frequency domain are a one-dimensional, complex function. However, the Laplace transform of the 1D signal is a complex function defined over a two-dimensional complex plane, called the s-plane, spanned by two variables, one for the horizontal real axis and one for the vertical imaginary axis. If this 2D function is evaluated along the imaginary axis, the Laplace transform simply becomes the Fourier transform.

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