![time domain corresponds to multiplication by the frequency,[note 1] so some differential
equations are easier to analyze in the frequency domain. Also, convolution in the time domain
corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any
linear time-invariant system, such as a filter applied to a signal, can be expressed relatively
simply as an operation on frequencies.[note 2] After performing the desired operations,
transformation of the result can be made back to the time domain. Harmonic analysis is the
systematic study of the relationship between the frequency and time domains, including the kinds
of functions or operations that are "simpler" in one or the other, and has deep connections to
almost all areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out
across the frequency domain and vice versa, a phenomenon known as the uncertainty principle.
The critical case for this principle is the Gaussian function, of substantial importance in
probability theory and statistics as well as in the study of physical phenomena exhibiting normal
distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian
function. Joseph Fourierintroduced the transform in his study of heat transfer, where Gaussian
functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an
integral transform, although this definition is not suitable for many applications requiring a more
sophisticated integration theory.[note 3] For example, many relatively simple applications use
the Dirac delta function, which can be treated formally as if it were a function, but the
justification requires a mathematically more sophisticated viewpoint.[1] The Fourier transform
can also be generalized to functions of several variables on Euclidean space, sending a function
of 3-dimensional space to a function of 3-dimensionalmomentum (or a function of space and
time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in
the study of waves, as well as in quantum mechanics, where it is important to be able to represent
wave solutions as functions of either space or momentum and sometimes both. In general,
functions to which Fourier methods are applicable are complex-valued, and possibly vector-
valued.[2] Still further generalization is possible to functions on groups, which, besides the
original Fourier transform on or n (viewed as groups under addition), notably includes the
discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group =
mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle closed
finite interval with endpoints identified). The latter is routinely employed to handle periodic
functions. The Fast Fourier transform (FFT) is an algorithm for computing the DFT.](https://image.slidesharecdn.com/inthistaskiwantyoutoverifythatthephaseresponseoffourie-230323082213-5bc9adbb/85/In-this-task-I-want-you-to-verify-that-the-phase-response-of-Fourie-pdf-2-320.jpg)
In this task, I want you to verify that the phase response of Fourie.pdf
- 1. In this task, I want you to verify that the phase response of Fourier transform is much more important than the magnitude response in case of images 1- Upload two images into Matlab of the same size. 2- Find the Fourier transform of these two images. (the function that you need to use is “y=fft2(im)” 3- Reconstruct the images using their corresponding Fourier transforms and verify that the Fourier transformation is perfectly invertable. (The function that you need to use is ifft2(y). 4- The Fourier transforms of the images are complex numbers, so you need to find their magnitude and phase. (The functions that you need to use are “abs(y)” for magnitude and “unwrap(angle(y))” for phase.). 5- Use the magnitude response of image 1 and the phase response of image 2 and reconstruct the image applying inverse Fourier transform. (The function that you need to use is ifft2(rcos+jrsin ), where r is the magnitude and is the phase). 6- Use the magnitude response of image 2 and the phase response of image 1 and reconstruct the image applying inverse Fourier transform. 7- Comment on the results. Solution Fourier transform :- The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time. Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the
- 2. time domain corresponds to multiplication by the frequency,[note 1] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, can be expressed relatively simply as an operation on frequencies.[note 2] After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to almost all areas of modern mathematics. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourierintroduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.[note 3] For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.[1] The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional space to a function of 3-dimensionalmomentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either space or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector- valued.[2] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on or n (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group = mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The Fast Fourier transform (FFT) is an algorithm for computing the DFT.