Orthogonality Property of Continuous Time Fourier Transform

Continuous Time Fourier Transform

Continuous time Fourier transform of x(t) is defined as X(ω)=∫−∞+∞x(t)e−jωtdt and discrete time Fourier transform of x(n) is defined as X(ω)=Σ∀nx(n)e−ωn.

From: Nonlinear Digital Filters , 2007

Discrete-Time Signal Processing

W. Kenneth Jenkins , ... Bill J. Hunsinger , in Reference Data for Engineers (Ninth Edition), 2002

Sampling and Reconstruction

The traditional CT Fourier transform can be applied to the ideal sampled signal, s*(t), as follows:

(Eq. 10) $\begin{array}{c}\Im \left\{s^{*}\left(t\right)\right\}=\sum _{n=-\infty }^{+\infty }s\left(nT\right)\Im \left\{{\delta }_a\left(t-nt\right)\right\}\\ =\sum _{n=-\infty }^{+\infty }{s}_a\left(nT\right)e^{-jnT\omega }\\ =\text{DTFT}\left\{s_a\left(nT\right)\right\}\end{array}$

This verifies that the traditional Fourier transform of s*(t) is identical to the DTFT of s(n), where s(n) ≡ s_a (t)| _{t = nT} ; i.e., the sequence s(n) is derived from s_a (t) by ideal sampling. This proves that s*(t) and s(n) are really different models of the same phenomenon, since their spectra (as computed with appropriate transforms) are identical. Suppose that S_a (jΩ) is the spectrum of s_a (t) and S(e ^jω ) is the spectrum of s(n). It can be shown that

(Eq. 11) $S\left(e^{j\omega }\right)=\left(1/T\right)\sum _{r=-\infty }^{+\infty }{S}_a\left(j\left[\Omega -2\pi r/T\right]\right)$

where ω = ΩT is often referred to as the "normalized digital frequency." Eq. 11 shows that the DT spectrum is formed from a superposition of an infinite number of replicas of the analog signal, as illustrated in Fig. 2. As long as the sampling frequency, Ω _s = 2π/T, is chosen so that Ω _s > 2Ω _B , where Ω _B is the highest frequency component contained in s_a (t), then each period of S(e ^jω ) contains a perfect copy of S_a (jΩ), and s_a (t) can be recovered exactly from s(n) by ideal low-pass filtering. Sampling under these conditions is said to satisfy the Nyquist sampling criterion, since the sampling frequency exceeds the Nyquist rate, 2Ω _B. If the sampling rate does not satisfy the Nyquist criterion, the adjacent periods of the analog spectrum will overlap, causing a distorted spectrum (see Fig. 2). This effect, called aliasing distortion, is rather serious because it cannot be easily corrected once it has occurred. In general, an analog signal should be prefiltered with an analog low-pass filter prior to sampling so that aliasing distortion does not occur.

Fig. 2. Relationship between the spectrum of an analog signal and the spectrum of the ideally sampled signal.

If the Nyquist criterion has been satisfied, it is always possible to reconstruct an analog signal from its samples according to

(Eq. 12) $s_a\left(t\right)=\sum _{k=-\infty }^{+\infty }{s}_a\left(kT\right)\sin \text{c}\left(\left(\pi /T\right)\left(t-kT\right)\right)$

This reconstruction formula results by filtering s*(t) with an ideal low-pass analog filter with a bandwidth of Ω _B = π/T (see Fig. 2). In general, exact reconstruction requires an infinite number of samples, although a good approximation can be obtained by using a large but finite number of terms in Eq. 12.

Most practical systems use a digital-to-analog converter for reconstruction, which results in an analog staircase approximation to the true analog signal; i.e.

(Eq. 13) ${\hat{S}}_a\left(t\right)=\sum _{k=-\infty }^{+\infty }{s}_a\left(kT\right)\left[u\left(t-kT\right)-u\left(t-\left(k+1\right)T\right)\right]$

It can be shown that Ŝ_a (t) is obtained by filtering s_a *(t) with an analog filter whose frequency response is

(Eq. 14) $H_a\left(j\Omega \right)=2T{e}^{-j\Omega T/2}\sin \text{c}\left(\Omega T/2\right)$

The approximation ${\hat{S}}_a\left(t\right)$ is said to contain "sin x/x distortion," which occurs because H_a (jΩ) is not an ideal low-pass filter. The H_a (jΩ) response distorts the signal by causing a droop near the band edge, as well as passing high-frequency distortion terms that "leak" through the side lobes of H_a (jΩ). Therefore, a practical D/A converter is normally followed by a postfilter

(Eq. 15) $H_p\left(j\Omega \right)=\left\{\begin{array}{c}{H}_a^{-1}\left(j\Omega \right),0\le \left|\Omega \right|\le \pi /T\\ 0,\Omega \text{otherwise}\end{array}\right\}$

which compensates for the distortion and produces the correct s_a (t) analog output. Notice, however, that H_p (jΩ) can only be approximated in practice, so that the best reconstruction is necessarily an approximation. Fig. 3 shows a digital processor complete with sampling and reconstruction devices at the input and output.

Fig. 3. Elements required for the digital processing of an analog signal.

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Finite Word Length Effects

Lars Wanhammar , in DSP Integrated Circuits, 1999

L ₂-Norm

The L ₂-norm of a continuous-time Fourier transform is related to the power contained in the signal, i.e., the rms value. The L ₂-norm of a discrete-time Fourier transform has an analogous interpretation. The L ₂-norm is

(5.8) $||X\left(e^{j\omega T}\right)||{}_2=\sqrt{\frac{1}{2\mathrm{\pi }}{\displaystyle {\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}|X\left(e^{j\omega T}\right)}|{}^2}d\omega T$

The L ₂-norm is simple to compute by using Parsevals relation [5, 27, 33], which states that the power can be expressed either in the time domain or in the frequency domain. We get from Parsevals relation

(5.9) $||X||{}_2=\sqrt{{\displaystyle \sum _{n=-\infty }^{\infty }x{\left(n\right)}^2}}$

We frequently need to evaluate L ₂-norms of frequency responses measured from the input of a filter to the critical overflow nodes. This can be done by implementing the filter on a general-purpose computer and using an impulse sequence as input. The L ₂-norm can be calculated by summing the squares of the signal values in the node of interest. This method of computing the L ₂-norm will be used in Example 5.6.

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REVIEWS

Wing-Kuen Ling , in Nonlinear Digital Filters, 2007

Definitions of continuous time Fourier transform, discrete time Fourier transform, and discrete Fourier transform

The Fourier analysis evaluates signals and systems in the frequency domain. Continuous time Fourier transform of x(t) is defined as $X(\omega )={\int }_{-\infty }^{+\infty }x(t)e^{-j\omega t}dt$ and discrete time Fourier transform of x(n) is defined as X(ω)=Σ_∀n x(n)e ^−ωn . It is worth noting that the discrete time Fourier transform is always 2π periodic, while this is not the case for the continuous time Fourier transform. Also, both the continuous time and discrete time Fourier transforms are defined in the frequency domain, which is a continuous domain. On the other hand, the discrete Fourier transform of x(n) is defined as $\overline{X}(k)=\frac{1}{N}{\sum }_{n=0}^{N-1}x(n)e^{-\frac{j2\pi nk}{N}}$ , in which the discrete Fourier transform is a map from an N point sequence to an N point sequence. The inverse continuous time Fourier transform of X(ω) is defined as $x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }X(\omega )e^{j\omega t}d\omega$ , the inverse discrete time Fourier transform of X(ω) is defined as $x(n)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(\omega )e^{j\omega n}d\omega$ , and the inverse discrete Fourier transform of $\tilde{X}(k)$ is defined as $x(n)={\sum }_{n=0}^{N-1}\overline{X}(k)e^{\frac{j2\pi nk}{N}}$ for n = 0, 1,…, N − 1. For a continuous time periodic signal with period T, X(ω) would consist of impulses located at $\frac{2\pi k}{T}\forall k\in Z$ . Hence, its time domain can be represented as continuous time Fourier series $x(t)={\sum }_{\forall k}{a}_k{e}^{\frac{j2\pi nk}{T}}$ , where $a_k=\frac{1}{T}{\int }_0^{T}x(t)e^{-\frac{jk\pi t}{T}}dt\forall k\in Z$ are called the continuous time Fourier coefficients. Similarly, for a discrete time periodic signal with period N, X(ω) would consist of impulses located at $\frac{2\pi k}{N}$ for k = 0, 1,…, N − 1 and 2π periodic elsewhere. Hence, its time domain can be represented as discrete time Fourier series $x(n)={\sum }_{n=0}^{N-1}{a}_k{e}^{\frac{j2\pi nk}{N}}$ , where $a_k=\frac{1}{N}{\sum }_{n=0}^{N-1}x(n)e^{-\frac{j2\pi nk}{N}}$ are called the discrete time Fourier coefficients. It is worth noting that there are exactly N discrete time Fourier coefficients, while this is not the case for the continuous time Fourier coefficients.

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Evoked Potentials

Leif Sörnmo , Pablo Laguna , in Bioelectrical Signal Processing in Cardiac and Neurological Applications, 2005

Shifts in continuous-time signals.

The influence of latency shifts on the ensemble average can be studied in terms of the earlier adopted "signal-plus-noise" model in (4.4), but modified to account for an unknown latency shift τ. Since τ is continuous-valued, we will consider the continuous-time counterpart to the model in (4.4),

(4.107) $x_i(t)=s(t-{\tau }_i)+{\upsilon }_i(t),$

where t denotes time and τ_i, i = 1, …,M are samples of the random variable T which is completely characterized by the PDF P_τ (τ). Based on the observation model in (4.107), the expected value of the ensemble average ŝ _a (t) is given by

(4.108) $\begin{array}{c}E[{\hat{s}}_a(t)]=\frac{1}{M}{\displaystyle \sum _{i=1}^M}E[s(t-{\tau }_i)]\\ ={\int }_{-\infty }^{\infty }s(t-\tau )p_{\tau }(\tau )d\tau .\end{array}$

Introducing the characteristic function of Pτ(τ) [62, p. 115],

(4.109) $P_{\tau }(\Omega )={\int }_{-\infty }^{\infty }{p}_{\tau }(\tau )e^{j\Omega \tau }d\tau ,$

the convolution integral in (4.108) can be expressed as a product in the frequency domain,

(4.110) $E[{\hat{S}}_a(\Omega )]=S(\Omega )P_{\tau }^{*}(\Omega ),$

where ŝ _a (Ω) and S(Ω) are the continuous-time Fourier transforms of ŝ _a (t) and s(t), respectively; Ω = 2πF where F denotes analog frequency and the asterisk (*) denotes the complex-conjugate. ⁷

In most cases of practical interest, the PDF p _τ(τ) can be assumed to be symmetric around τ = 0, with tails that decrease monotonically to zero. As a result, the effect of P _τ(Ω) on the original signal S(Ω) in (4.110) is equivalent to filtering of s(t) with a linear, time-invariant, lowpass filter whose impulse response is given by P _τ(τ). The ensemble average computed in the presence of latency shifts is thus biased and will not approach s(t) as the number of EPs increases.

An example of P _τ(τ) is the zero-mean, Gaussian PDF with a variance σ ² _τ whose characteristic function is

This type of latency shift acts as a lowpass filter on s(t) to a degree determined by σ _τ, It is of particular interest to compute the −3 dB cut-off frequency, denoted F_c, as a function of σ_τ :

$e^{-2{(\pi F_c{\sigma }_{\tau })}^2}=\frac{1}{\sqrt{2}},$

or

Figure 4.20(a) displays the relationship between F_c and σ _τ For example, it can be seen that a dispersion of σ _τ = 1 ms corresponds to lowpass filtering with a cut-off frequency of 133 Hz.

Figure 4.20. : The −3 dB cut-off frequency F_c associated with the lowpass filtering effect due to latency shifts; F_c is plotted as a function of (a) the standard deviation σ_τ of a Gaussian PDF and (b) the sampling interval T of a uniform PDF.

Another type of latency shift is that caused by sampling ("sampling jitter"), typically assumed to have a uniform PDF over the sampling interval T,

(4.113) $p_{\tau }(\tau )=\{\begin{array}{ll}\frac{1}{T},\hfill & -T/2\le \tau \le T/2;\hfill \\ 0,\hfill & \text{otherwise}\text{.}\hfill \end{array}$

The lowpass filtering effect due to sampling is described by the corresponding sine characteristic function

(4.114) $P_{\tau }(\Omega )=\frac{sin\frac{1}{2}\Omega T}{\frac{1}{2}\Omega T}.$

In contrast to the Gaussian case in (4.112), it is difficult to derive a closed-form expression from (4.114) relating Fc to the dispersion parameter T. The desired relationship is, however, easily calculated by numerical techniques and is presented in Figure 4.20(b).

When _Pτ (τ) is known, the influence of latency shifts on the ensemble average can be determined using (4.110). However, it is actually possible to obtain certain information on the statistics of τ without knowledge of the PDF. For example, the variance of τ can be estimated from the ensemble variance [63].

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Advanced signal processing techniques for feature extraction

Md Shafiullah , ... A.H. Al-Mohammed , in Power System Fault Diagnosis, 2022

4.2 Signal processing techniques

Among many signal-processing techniques, the FT is the most widely used one for transforming the time-domain signals into the frequency domain signals. It retrieves the global frequency content of the stationary signals. The continuous-time FT deals with continuous-time signals. In contrast, the discrete Fourier transform (DFT) derives the frequency-domain (spectral) representations of the finite discrete signals in the time domain. In the case of many samples, the DFT becomes computationally expensive. Also, DFT suffers from aliasing and leakage problems. However, the fast Fourier transform, an efficient and easy to implement algorithm, is employed to reduce the DFT computation burden. However, it cannot effectively deal with the nonstationary signals and loses temporal information. Besides, it is sensitive to noise that requires additional filtering [1,32]. Dennis Gabor proposed the short-time Fourier transform (STFT) to adapt the FT to analyze the nonstationary signals. The STFT uses a small sampling window of the regular interval and decomposes the signals into the frequency domain. However, it creates a resolution problem between frequency and time. A good frequency resolution may result in poor time resolution and vice versa [1,14,32,33]. To overcome the resolution issue, the researchers introduce advanced SPT as WT, ST, and filter banks [4].

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Frequency-domain representation of discrete-time signals

Edmund Lai PhD, BEng , in Practical Digital Signal Processing, 2003

4.2 Discrete Fourier transform for discrete-time aperiodic signals

When a discrete-time signal or sequence is non-periodic (or aperiodic), we cannot use the discrete Fourier series to represent it. Instead, the discrete Fourier transform (DFT) has to be used for representing the signal in the frequency domain. The DFT is the discrete-time equivalent of the (continuous-time) Fourier transforms. As with the discrete Fourier series, the DFT produces a set of coefficients, which are sampled values of the frequency spectrum at regular intervals. The number of samples obtained depends on the number of samples in the time sequence.

A time sequence x(n) is transformed into a sequence X(ω) by the discrete Fourier transform.

(3) $X\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)e^{-\text{j}2\pi \text{k}\text{n}/\text{N}}k=0,1,\dots ,N-1$

This formula defines an N-point DFT. The sequence X(k) are sampled values of the continuous frequency spectrum of x(n). For the sake of convenience, equation 3 is usually written in the form

(4) $X\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)W_{\text{N}}^{\text{k}\text{n}}k=0,1,\dots ,N-1$

where

Note that, in general, the computation of each coefficient X(k) requires a complex summation of N complex multiplications.

Since there are N coefficients to be computed for each DFT, a total of N ² complex additions and N ² complex multiplications are needed. Even for moderate values of N, say 32; the computational burden is still very heavy. Fortunately, more efficient algorithms than direct computation are available. They are generally classified, as fast Fourier transform algorithms and some typical ones will be described later in the chapter.

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Vibration based condition monitoring and fault diagnosis of wind turbine planetary gearbox: A review

Tianyang Wang , ... Zhipeng Feng , in Mechanical Systems and Signal Processing, 2019

2.1 Basic analysis of the spectral complexity and transfer path effect

The complex structure of the WT planetary gearbox can be directly reflected on the complexity of the spectrum. Understanding the spectrum, especially the sideband pattern around the harmonics of the meshing frequency is of great importance for the WT planetary gearbox fault diagnostics. For this, several researchers have made a large amount of contribution on this sub-topic with the signal simulation and dynamic model.

The signal simulation model is a kind of mathematical models which can simulate the main characteristic features directly. Researchers can deduce the theoretical spectrum construction based on this kind of models. In early times, several researchers noticed that the spectrum of the planetary gearbox vibration response is different from the one of fixed – axis gearbox with asymmetric sidebands around the meshing frequency. This abnormal phenomenon has attracted much attention. At very first, McFadden and Smith [39] observed that the sidebands around the tooth meshing frequency are not symmetrical and the prominent spectral peak is generally the sideband rather than the meshing frequency and gave the explanation based on a model of the vibration transmission It is stated that the asymmetry is caused by the fixed sensor and rotating carrier, not a substantive feature. Based on the same model, McNames [40] analyzed and explained the asymmetrical phenomenon through the continuous-time Fourier transform, which can forecast the relative amplitudes of the dominant peaks around the meshing frequency and its harmonics.

Similarly, Mosher [41] analyzed the corresponding vibration spectra with a kinematic model for fault detection. The frequencies with higher amplitudes around the harmonics of meshing frequency can be predicted. It should be noted that these researches are mainly to explain the underlying reasons for the asymmetric sidebands and the corresponding distribution pattern. The constructed model is relatively simple, which cannot reflect the real situation.

To consider the real situations comprehensively, Inalpolat and Karhraman (2009) built a simplified mathematical model to explain the mechanisms of planetary gear sets modulation sidebands. The proposed model is more general for different planet spacing and meshing phasing conditions. The planetary gear set is classified into five distinct groups based on their sideband patterns [42]. A similar study proposed by Vicuña [43] divided the planetary gearboxes into four types according to the spectral structure. Inalpolat and Karhraman (2010) predicted the modulated sidebands pattern considering the manufacturing errors using a dynamic model [44]. Luo et al. [45] explained the amplitude modulation of sequentially phased WT planetary gears with a mathematical method. Although these models can predict the spectrum more accurately, they do not consider the fault condition.

Different from the signal simulation model, a dynamic model is much more similar to the real vibration response. More accurate features can be obtained from this kind of model. For example, Karray et al. [46] built a torsional model to analyze the modulation sidebands of the planetary gear set. Liu et al. (2016) built a two-dimensional lumped-parameter dynamic model of a planetary gear set considering two transfer paths (inside path and casing path) [47]. Parra and Vicuña (2017) separately built a phenomenological model and lumped-parameter model to study the frequency characteristic of planetary gearbox vibrations under health and fault conditions. Moreover, a particular function is proposed to decompose the lumped-parameter model based simulated signal to the one with fixed reference[48]. Liu et al. (2018) developed a resultant vibration signal model of a single-stage planetary gear train considering the effects of the transmission path and the direction variation of the excitation source [49].

Recently, several researchers have focused their attention on how the time-varying vibration transfer paths affect the corresponding spectrum. Towards this problem, Vicuña (2012) distinguished the external meshing processes from the internal meshing processes. It stated that the former one could influence all the spectral components, the later can only affect the gear meshing frequency and it's harmonic [43]. Lei et al. (2016) built a phenomenological model of planetary gearboxes considering all the six possible transfer paths and the planet gear angular shift [50]. He et al. (2017) developed a mathematical model to analyze the healthy planetary gear train response considering the meshing vibrations of the planet–ring and planet–sun gear pairs, the time-varying transmission path and meshing force direction [51].

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