Relations between Fourier transforms and Fourier series

Introduction

Fourier Transform has very precise relations with the Fourier series, the DTFT and the DFT. The Fourier transform can be applied to time-discrete or time-periodic signals using the δ-Dirac formalism. In fact the Fourier series, the DTFT and the DFT can be derived all from the general Fourier transform, and are, from a theorical point of view, only particular cases of the Fourier transform.

In signal theory and DSP the DFT (implemented as FFT) is quite extensively used to calculate approximations of the spectrum of a continuous signal, knowing only a sequence of sampled point. The relations between DFT and Fourier transform are in this case essential.

Definitions

In the following table the definitions for the Continuous Fourier transform, Fourier series, DTFT and DFT are reported:

Fourier Transformations Definitions
×	Continuous Time	Discrete Time
Time aperiodic	$x(t)=\int _{-\infty }^{\infty }X(f)\ e^{i2\pi ft}\,df$	$x[n]=T\int _{-{\frac {1}{2T}}}^{\frac {1}{2T}}{\bar {X}}(f)\ e^{i2\pi fnT}\ dt$
-	$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\,dt$	${\bar {X}}(f)=\sum _{n=-\infty }^{+\infty }x[n]\ e^{-i2\pi fnT}\ df$
Time periodic	${\bar {x}}(t)=\sum _{k=-\infty }^{+\infty }X_{k}e^{i2\pi kf_{0}t}$	$x_{n}={\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}e^{i{\frac {2\pi }{N}}kn}\qquad n=0,\dots ,N-1.$
-	$X[k]={\frac {1}{T_{0}}}\int _{-{\frac {T_{0}}{2}}}^{\frac {T_{0}}{2}}{\bar {x}}(t)\;e^{-i2\pi kf_{0}t}$	$X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {2\pi i}{N}}kn}\qquad k=0,\dots ,N-1$

The table shows the properties of the time-domain signal:

Continuous time vs Discrete Time (columns),

Aperiodic in time vs Periodic in time (rows).

Equations needed to relate the various transformations

The definitions given in the previous section can be introduced axiomatically or can be derived from the Continuous Fourier transform using the extend formalism of Dirac delta. Using this formalism the Continuous Fourier transform can be applied also to discrete or periodic signals.

To calculate the continuous Fourier transform of discrete and/or periodic signals we need to use some equations called Poisson formulas and the properties of the Continuous Fourier transform. Here is reported a list of them:

1. The Poisson formulas:

\sum _{n=-\infty }^{+\infty }x(t-nT_{0})={\frac {1}{T_{0}}}\sum _{k=-\infty }^{+\infty }X\!\left({\frac {k}{T_{0}}}\right)\;e^{-j2\pi kf_{0}t}\qquad {\text{first Poisson formula}}

\sum _{n=-\infty }^{+\infty }x(nT)\;e^{-j2\pi nfT}={\frac {1}{T}}\sum _{k=-\infty }^{+\infty }X\!\left(f-{\frac {k}{T}}\right)\qquad {\text{second Poisson formula}}

2. The comb transform is important to understand the link between the continuous and the discrete or periodic case:

\sum _{k=-\infty }^{+\infty }\!\delta (t-kT)\quad \Longleftrightarrow \quad \sum _{i=-\infty }^{+\infty }\delta \!\left(f-{\frac {i}{T}}\right)

3. The last piece of information needed are the theorems which define the Fourier transform properties (in particular the convolution property).

All these equations and properties can be demonstrated on their own.

Once calculated, the continuous Fourier transform of discrete and/or periodic signals can be related to the DTFT, Fourier series ant to the DFT definitions given above.

Relationship between the various transform

The following figure represent the relations between the various transforms.

File:Fourier-rel-white-bg.png

Figure 2. A "cube" graph representing the relations between Fourier transform, DTFT, Fourier series and DFT. Each side of the cube indicates the operations needed to pass from one vertex to the other. The bolder double arrows indicate the link between each function in the direct domain and its transform.

DFT vs Continuous Fourier transform

Note that following the Poisson formulas we would obtain for the DFT the definition ${\tilde {X}}_{k}\,$ . However the DFT is defined usually as $X_{k}\,$ (see Figure 2 or the previous definitions). For this reason the link between the DFT and the periodical transform ${\bar {X}}(f)\,$ is different by a scale factor from the relation obtained by the application of the Poisson formulas (which bring to ${\tilde {X}}_{k}\,$ and not to $X_{k}\,$ ).

Samples of the spectrum of a sampled continuous signal can be accurately calculated if the signal is band-limited and the sampling is done at a frequency above the Nyquist frequency. In this case, if the signal is time limited, we can begin sampling it before the signal "begins" and stop sampling after the signal "ends". Calculating the DFT of this finite sequence obtained from such sampling we obtain the sampled values of the spectrum of the original signal, apart a scale factor $1/T$ (where T is the sampling step):

$\underbrace {X_{k}} _{DFT}=\underbrace {{\bar {X}}\left({\frac {k}{NT}}\right)} _{\text{DTFT}}={\frac {1}{T}}\sum _{i=-\infty }^{+\infty }\underbrace {X\left({\frac {k-iN}{NT}}\right)} _{\text{FT}}\simeq {\frac {1}{T}}\underbrace {X\left({\frac {k}{NT}}\right)} _{\text{FT}}$

The last $\simeq$ equality is between the spectrum of the continuous signal and the periocized spectrum in one spectral period. The $\simeq$ symbol is also is used to stress that if the signal is not perfectly band limited we always get a bit of aliasing so so the equality is not exact.

DTFT vs Continuous Fourier transform

to be written

Fourier series vs Continuous Fourier transform