Relations between Fourier transforms and Fourier series

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In the mathematical field of harmonic analysis, the continuous Fourier transform has very precise relations with Fourier series. It is also closely related to the discrete-time Fourier transform (DTFT) and the discrete Fourier transform (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 continuous Fourier transform. They are, from a theoretical point of view, particular cases of the Fourier transform.

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

Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
Related transforms

Contents

[edit] 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^{ i 2 \pi f t}\,df x[n] = T \int_{-\frac{1}{2T}}^{\frac{1}{2T}} \bar{X}(f) \ e^{ i2\pi fnT} \ df
- X(f) = \int_{-\infty}^{\infty} x(t)\ e^{-i 2\pi f t}\,dt \bar{X}(f) = \sum_{n=-\infty}^{+\infty} x[n]\ e^{-i2\pi fnT}
Time periodic \bar{x}(t) = \sum_{k=-\infty}^{+\infty}\! X[k]\; e^{i2\pi k f_0 t} x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \; e^{i\frac{2\pi}{N} k n} \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_0t}\,dt X_k = \sum_{n=0}^{N-1} x_n \; e^{-i\frac{2 \pi}{N} k n} \qquad k = 0, \dots, N-1

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

  • Continuous time versus Discrete Time (columns),
  • Aperiodic in time versus Periodic in time (rows).
Figure 1. Definitions of the various Fourier transform and their inverse. Starting from the upper left side and going clock-work we have: Fourier transform, DTFT, DFT and Fourier series
Figure 1. Definitions of the various Fourier transform and their inverse. Starting from the upper left side and going clock-work we have: Fourier transform, DTFT, DFT and Fourier series

[edit] 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 introduce some equations and recall some Fourier transform properties. Here is reported a list of them:

1. The first Poisson summation formula:

\sum_{n=-\infty}^{+\infty} x(t - nT_0) = \frac{1}{T_0} \sum_{k=-\infty}^{+\infty} X\!\left(\frac{k}{T_0}\right) \; e^{i2\pi kf_0 t}

2. The second Poisson summation formula:

\sum_{n=-\infty}^{+\infty} x(nT) \; e^{-i2\pi n f T} = \frac{1}{T} \sum_{k=-\infty}^{+\infty} X\!\left(f-\frac{k}{T}\right)

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

\sum_{n=-\infty}^{+\infty}\!\delta(t - nT) \quad\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{T} \sum_{k=-\infty}^{+\infty}\delta \!\left( f -\frac{k}{T} \right)

4. 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, the Fourier series and to the DFT definitions given above.

[edit] Relationship between the various transform

The following figure represent the relations between the various transforms.

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.
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.

Explanation of the symbols:

  • The signal and its transform are bound by a bold double arrow (\leftrightarrow)
  • x[n]\, and X[k]\, are infinite sequences
  • \bar{x}(t) and \bar{X}(f) are periodic functions
  • x_n\,, X_k\, and \tilde{X}_k are finite sequences
  • \mathcal{F}\{\dots\} indicates exclusively the continuous Fourier transform.

The Poisson summation formulas allow to link the Fourier series and the DTFT to the Fourier transform (respectively formula 1. and 2.).

The convolution property (4.) and Dirac comb transform (3.) allow to calculate the Fourier transform for the time-periodic or time-discrete signals as function of X(f)\,. In Figure 2 is showed what operations correspond, in the spectral domain, to the sampling of a continuous signal or to the periodicization of an aperiodic signal.

From Figure 2 we can see that the time domain sampling has the same effect on the spectrum both for an aperiodic signal (x(t)) and for a periodic signal (\bar{x}(t)). Conversely, the time domain periodicization has he same spectral effect on a continuous signal (x(t)) and on a discrete signal (x[n]).

[edit] DFT versus continuous Fourier transform

The discrete Fourier transform (DFT) is the transform of a finite sequence. A finite sequence can be thought of as a time-periodic and time-discrete signal considered only in one period. For this reason the spectrum must be both periodic and discrete.

Following the Poisson formulas we would obtain \tilde{X}_k\, as DFT definition. 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\,).

Sample points of the spectrum of a 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}} \qquad k = 0, \dots, N-1

The last \simeq equality is between the periodic spectrum \bar{X}(f) evaluated in one period and the spectrum of the continuous signal X(f). The \simeq symbol is also used to stress that, if the signal is not perfectly band limited, we always get a bit of aliasing so the equality is not exact.

Usually in digital signal processing (DSP) the signal is too long to be analyzed as a whole. In this case windowing is used to calculate approximate spectrum samples of a small portion of the entire signal. This process inevitably adds further errors such leakage and scalloping loss (see Window function).

[edit] DTFT versus continuous Fourier transform

The Discrete Time Fourier Transform (DTFT) is the transform of a discrete sequence. Since the time-domain is discrete the spectrum is periodic.

A discrete signal x[n] can thought as the sampling of a continuous signal x(t) with step T. The sampled signal can be treated as a continuous signal using the Dirac delta formalism. In particular the sampling operation is equivalent to the multiplication by a Dirac comb:

\begin{align} x_{\rm sampled}(t) &= x(t) \cdot \left( \sum_{n=-\infty}^{+\infty}\!\delta(t - nT) \right) = \sum_{n=-\infty}^{+\infty} \! x(t) \; \delta(t - nT) = \\ &= \sum_{n=-\infty}^{+\infty} \! x(nT) \; \delta(t - nT) = \sum_{n=-\infty}^{+\infty} \! x[n] \; \delta(t - nT) \end{align}

Calculating the Fourier transform of the sampled signal using the convolution property (3.) and the comb transform (2.), and then applying the second Poisson summation formula, we obtain:

\mathcal{F}\{x_{\rm sampled}(t)\} = \frac{1}{T} \sum_{k=-\infty}^{+\infty} X\!\left(f-\frac{k}{T}\right) = \sum_{n=-\infty}^{+\infty} x(nT) \; e^{-i2\pi n f T} = \bar{X}(f)

where X(f) is the Fourier transform of the continuous signal x(t). We see that the Fourier transform of xsampled(t) is equal to the DTFT of x[n]. The DTFT definition can be seen as formula to calculate the Fourier transform of the sampled signal using only the sampled values x[n] (without the Dirac delta formalism). The last equation is reported in the lower left corner of Figure 2.

Another important aspect to note is that the time-domain sampling with step T corresponds to a periodicization of the spectrum with period 1 / T and a multiplication of the spectrum by an 1 / T factor. This relation can be seen in Figure 2 along the vertical arrows that go from x(t) to x[n] and from X(f) to \bar{X}(f).

[edit] Fourier series versus continuous Fourier transform

The Fourier series is an expansion of a periodic signal as a linear combination of discrete harmonic components. Since the signal is time-periodic the spectral components are not spread over a continuum range of frequency but are concentrated in discrete, equally spaced, frequency values. These discrete frequencies are all multiple of a base harmonic called fundamental. The fundamental harmonic is equal to the inverse of the period of the signal.

A periodic signal \bar{x}(t) can thought as the periodicization with period T0 of an aperiodic signal x(t). In particular, the periodicization is equivalent to the convolution (*\, symbol) of x(t) by a Dirac comb:

x_{\rm periodic}(t) = \bar{x}(t) = x(t) \; * \sum_{n=-\infty}^{+\infty}\!\delta(t - nT_0) = \sum_{n=-\infty}^{+\infty} x(t - nT_0)

Calculating the Fourier transform of the periodic signal using the convolution property (4.) and the comb transform (3.), and then applying the first Poisson summation formula (1.), we obtain:

\begin{align} \mathcal{F}\{x_{\rm periodic}\} \;\; &\stackrel{3,4}{=} \;\; X(f) \cdot \frac{1}{T}\sum_{k=-\infty}^{+\infty}\delta \!\left( f -\frac{k}{T} \right) = \frac{1}{T}\sum_{k=-\infty}^{+\infty} X\! \left( \frac{k}{T} \right) \; \delta \!\left( f -\frac{k}{T} \right) = \\ &\stackrel{1}{=} \sum_{k=-\infty}^{+\infty} \! X[k] \; \delta \!\left( f -\frac{k}{T} \right) \end{align}

where X(f) is the Fourier transform of the aperiodic signal x(t), and X[k] are the coefficients of the Fourier series expansion for the periodic signal \bar{x}(t). This equation shows that the coefficients of the Fourier expansion of a periodic signal are equal to the amplitudes of the Dirac deltas of the Fourier transform. The last equation is reported in the upper right corner of Figure 2.

Another important aspect is that the time-domain periodicization with period T0 corresponds, in the frequency domain, to a discretization (sampling) of the spectrum with step 1 / T0 and to a multiplication by a 1 / T0 factor. This relation can be seen in Figure 2 along the horizontal arrows that go from x(t) to \bar{x}(t) and from X(f) to X[k].

[edit] See also

[edit] References

  • M. Luise, G. M. Vitetta: Teoria dei segnali, MacGraw-Hill, ISBN 88-386-0809-1 (Italian version only)
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