# 数学代写|傅里叶分析代写Fourier analysis代考|MAST20026

## 数学代写|傅里叶分析代写Fourier analysis代考|Complex Data

Consideration of complex data is beyond the scope of this book but, in principle, if we were to perform a real Fourier series decomposition of complex data then we would obtain separate real Fourier series for the real and imaginary parts. We could then combine these two series term-wise according to frequency into a single series having real and imaginary parts for each frequency term.

Also, if we were to perform a complex Fourier series decomposition of complex data then we would obtain separate series of complex exponential terms for the real and imaginary parts. We could then combine these two series term-wise according to frequency into a single series having real and imaginary parts for each frequency term.

In terms of the symmetries of positive and negative frequency components, for complex data the situation is similar to that for real data but more complicated as we also have to consider the conjugate-symmetries of the imaginary part. These can be considered as a mirror-image of the conjugatesymmetries for the real part. However, we would still be able (at least in principle) to obtain all the real sinusoidal coefficients from the complex coefficients and back-substitute from the series of complex exponentials to obtain real Fourier series for the real and imaginary parts.

## 数学代写|傅里叶分析代写Fourier analysis代考|Frequency Functions

If we wished, we could construct an explicit frequency function, $X(f)$, from the Fourier series coefficients for a time function, $x(t)$, and this would be a discrete frequency-domain representation of $x(t)$. We would then have a relationship between the time and frequency domains in the sense that we could completely represent any time function as a frequency function by making use of its harmonic composition.

Starting with the time function, i.e. real Fourier series as in Equations 3.5, we have
$$x(t)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left(a_k \cos \left(2 \pi f_k t\right)+b_k \sin \left(2 \pi f_k t\right)\right)$$
or, for complex Fourier series, Equation 3.20, we have
$$x(t)=\sum_{k=-\infty}^{\infty} c_k e^{2 \pi i f_k t} .$$

We know both the frequencies for $k=1,2,3, \ldots$ and the forms that the terms in the summation must take, which means that the frequency function can be simplified to the ordered (indexed) list of coefficients, i.e. $\left{\left(a_k, b_k\right)\right}$ or $\left{c_k\right}$ for real and complex Fourier series respectively. More completely, the frequency function is, in terms of the real Fourier series coefficients
$$X(f)=\frac{a_0}{2}+\left{\left(a_k, b_k\right)\right}_{k=1}^{\infty}$$
or, in terms of the complex Fourier series coefficients
$$X(f)=\left{c_k\right}_{k=-\infty}^{\infty} .$$
If we know all the coefficients then we can completely reconstruct the original function, $x(t)$, and so $X(f)$ is a frequency-domain representation of $x(t)$ according to the Fourier series formalisation.

Both forms in Equations $4.3$ and $4.4$ are discrete frequency functions because the finite duration of the time function, $T$, restricts the lowest nonzero frequency we can resolve to $f_1=1 / T$ and higher frequencies are restricted to integer multiples (harmonics) of this fundamental frequency because of the orthogonality conditions (Equations 3.4).

We will start from here in our derivation of the discrete Fourier transform (DFT) but, first, we will briefly consider the (analytical or continuous-time) Fourier transform to highlight some important general features of Fourier spectra.

## 数学代写|傅里叶分析代写Fourier analysis代考|Frequency Functions

$$x(t)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left(a_k \cos \left(2 \pi f_k t\right)+b_k \sin \left(2 \pi f_k t\right)\right)$$

$$x(t)=\sum_{k=-\infty}^{\infty} c_k e^{2 \pi i f_k t} .$$

$$X(f)=\backslash \text { left }{c / k \backslash \text { right }}_{-}{k=-\backslash i n f t y}^{\wedge}{\text { infty }} \text { 。 }$$

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