how to find impulse response of discrete system

In the digital age, any medical image needs to be transformed from continuous domain to discrete domain (i. eastward. one'due south and 0's) in society to be represented in a computer. To exercise and so, we have to understand what a continuous and a detached point is.

In the digital historic period, any medical epitome needs to exist transformed from continuous domain to discrete domain (i.due east. one'south and 0's) in order to be represented in a computer. To practice so, we have to sympathize what a continuous and a discrete signal is. Both of them are handled by systems which will also be introduced in this chapter. Another primal concept is the Fourier transform as it allows us to represent any time domain signal in frequency space. In particular, we will find that both representations – fourth dimension domain and frequency domain – are equivalent and can be converted into each other. Having found this important relationship, we can then determine weather which volition guarantee that besides conversion from continuous to discrete domain and vice versa is possible without loss of data. On the way, we will innovate several other important concepts that will besides observe repeated use later in this volume.

2.i. Signals and Systems

2.i.i. Signals

A bespeak is a part f (t) that represents information. Often, the independent variable t is a concrete dimension, like time or space. The output f of the signal is too called the dependent variable. Signals are everywhere in everyday life, although nosotros are mostly not enlightened of them. A very prominent instance is the speech signal, where the contained variable is time. The dependent variable is the electric signal that is created by measuring the changes of air pressure using a microphone. The clarification of the speech generation process enables to practise efficient speech processing, eastward.yard., radio transmission, oral communication coding, denoising, speech recognition, and many more. In general, many domains can exist described using organization theory, due east.thousand., biological science, gild, economic system. For our application, nosotros are mainly interested in medical signals.

Both the dependent and the independent variable can be multidimensional. Multidimensional independent variables t are very common in images. In normal camera images, space is described using ii spatial coordinates. All the same, medical images, e.thou., CT volume scans, can also accept three spatial dimensions. Information technology is not necessary that all dimensions have the same significant. Videos have two spatial coordinates and one time coordinate. In the medical domain, we can also detect college-dimensional examples like time-resolved 4-D MR and CT with three spatial dimensions and one fourth dimension dimension. To represent multidimensional values, i.e., vectors, nosotros use bold-face up letters t or multiple scalar values, e.thousand., t = (10, y, z) ^T . The medical field as well contains examples of multidimensional dependent variables f. An case with many dimensions is the Electroencephalography (EEG). Electrodes are attached to the skull and measure electrical encephalon activity from multiple positions over time. To represent multidimensional dependent variables, nosotros likewise use boldface letters f .

The signals described in a higher place are all in continuous domain, e.g., time and space change continuously. As well, the dependent variables vary continuously in principle, like light intensity and electrical voltage. However, some signals be naturally in discrete domains w.r.t. the independent variable or the dependent variable. An example for a discrete betoken in dependent and independent variable is the number of first semester students in medical engineering science. The independent variable time is discrete in this instance. The starting semesters are WS 2009, WS 2010, WS 2011, and so on. Other points in time are considered to be abiding in this interval. The number of students is restricted to natural numbers. In general, it is likewise possible that just the dependent or the contained variable is discrete and the other one continuous. In addition to signals that are discrete by nature, other signals must be represented discretely for processing with a digital computer, which means that the independent variable must exist discretized earlier processing with a figurer. Furthermore, data storage in computers has limited precision, which means that the dependent variable must exist detached. Both are a straight issue of the finite memory and processing speed of computers. This is the reason why discrete arrangement theory is very important in exercise.

Signals can be further categorized into deterministic and stochastic signals. For a deterministic signal, the whole waveform is known and can be written down as a office. In contrast, stochastic signals depend randomly on the independent variable, e.m., if the bespeak is corrupted past noise. Therefore, for practical applications, the stochastic backdrop of signals are very important. However, deterministic signals are important to analyze the behavior of systems. A short introduction into stochastic signals and randomness will be given in Sec. 2.4.3.

This chapter is presents basic knowledge on how to correspond, analyze, and process signals. The correct processing of signals requires some math and theory. A more in-depth introduction into the concepts presented here can be institute in [3]. The awarding to medical information is treated in [2].

2.1.2. Systems

Signals are processed in processes or devices, which are abstracted equally systems. This includes not only technical devices, but natural processes like attenuation and reverberation of speech in transmission through air as well. Systems take signals as input and as output. Inside the organisation, the properties of the point are inverse or signals are related to each other. We describe the processing of a indicate using a organisation with the operator $\mathcal{H}\{.\}$ that is applied to the function f. A graphical representation of a organisation is shown in Fig. 2.ane.

Figure 2.1

A arrangement H{.} with the input signal f(t) and the output signal g(t).

An important subtype is the linear shift-invariant system. Linear shift-invariant systems are characterized by the two important properties of linearity and shift-invariance (cf. Geek Box ii.1 and 2.2).

Another property important for the practical realization of linear shift-invariant systems is causality. A causal system does not react to the input

Geek Box 2.one Linear Systems

The linearity property of a organisation means that linear combinations of inputs can be represented as the same linear combination of the processed inputs

$$\mathcal{H}\{f(t)+\mathrm{thou}(t)\}=\mathcal{H}\{f(t)\}+\mathcal{H}\{\mathrm{grand}(t)\},$$

with abiding a and capricious signals f and g. The linearity property greatly simplifies the mathematical and applied handling, as the behavior of the organization can be studied on basic signals. The behavior on more than complex signals tin be inferred directly if they tin can be represented as a superposition of the bones signals.

Geek Box two.two Shift-Invariant Systems

Shift-invariance denotes the characteristic of a system that its response is independent of shifts of the independent variable of the signal. Mathematically, this is described equally

for the shift τ. This means that shifting the betoken by τ followed by processing with the system is identical to processing the signal with the system followed by a shift with τ.

before the input actually arrives in the system. This is especially important for signals with time as the independent parameter. However, non-causal systems do not pose a problem for the contained parameter space, due east.g., paradigm filters that use data from the left and right of a pixel. Geek Box 2.3 presents examples for the combination of different system backdrop.

Linear shift-invariant systems are of import in practice and take convenient properties and a rich theory. For linear shift-invariant systems, the abstract operator $\mathcal{H}\{.\}$ can be described completely using the impulse response h (t) (cf. Sec. ii.2.2) or transfer function H (ξ) (cf. Sec. ii.iii.2). The impulse response is combined with the signal by the functioning of convolution. This is sufficient to depict all linear shift-invariant systems.

Geek Box ii.3 Organization Examples

Here are some examples of dissimilar systems analyzed w. r. t. linearity, shift-invariance, and causality. f(t) represents the input and g(t) the output signal.

• k(t) = 10f(t): linear, shift-invariant, causal

• 1000(t) = sin(f(t)): non-linear, shift-invariant, causal

• yard(t) = threef(t + ii): linear, shift-invariant, not-causal

• g(t) = f(t) − 2f(t − one): linear, shift-invariant, causal

• g(t) = f(t) · e^(−0.5t): linear, not shift-invariant, causal

2.2. Convolution and Correlation

This department describes the combination of signals in linear-shift-invariant systems, i. e., convolution or correlation. Earlier discussing indicate processing in particular, we will outset commencement by revisiting important mathematical concepts that will be needed in the following capacity.

2.2.1. Complex Numbers

Complex numbers are an extension to real numbers. They are defined as z = a + bi. a is called the existent function of z and b the imaginary office. Both human action as coordinates in a 2-D space. i is the imaginary unit that spans the 2d dimension of this space. The special meaning of i is that i ² = one. This makes complex numbers important for many areas in mathematics, − but also in many applied fields like physics and electric technology. To extract the coordinates of the complex number, we apply the post-obit definitions

Nosotros can directly write z = Re (z) + Im (z) i. Another important definition is the circuitous conjugate $\bar{z}$ , which is the aforementioned number as z except with the opposite sign for the imaginary part $\bar{z}$ = a − bi.

Real numbers are the subset of the complex − numbers for which b = 0, i. due east., no imaginary part. Geometrically, this ways that real numbers are defined on a one-dimensional centrality, whereas the complex numbers are defined on a 2-D aeroplane. The geometric interpretation of complex numbers is too helpful to run into the equivalence of the Cartesian coordinate notation z = a + bi and the polar coordinate notation z = A (cos ϕ + i sin ϕ) of circuitous numbers. The

Geek Box two.4 Complex Numbers and Geometric Interpretation

If a point on the 2-D airplane is seen as a position vector, A is the length of the vector and ϕ the bending relative to the existent axis. The 2 notations can be converted to each other using the following formulas:

$$\begin{array}{c}A=\sqrt{a^2+b^{\mathrm{two}}}\\ \varphi =\{\begin{array}{ll}\arctan \frac{b}{a},\hfill & \text{ if}a>0\hfill \\ \arctan \frac{b}{a}+\pi ,\hfill & \text{ if}a<0\text{ and}b\ge 0\hfill \\ \arctan \frac{b}{a}-\pi ,\hfill & \text{ if}a<0\text{ and}b<0\hfill \\ \frac{\pi }{2},\hfill & \text{ if}a=0\text{ and}b>0\hfill \\ -\frac{\pi }{2},\hfill & \text{ if}a=0\text{ and}b<0\hfill \\ \text{ undefined,}\hfill & \text{ if}a=0\text{ and}b=0\hfill \end{array}\\ a=A\cos \varphi \\ b=A\sin \varphi \end{array}$$

polar coordinates consists of magnitude A and bending ϕ (cf. Geek Box ii.4). For system theory, an important property of complex numbers is Euler'south formula

$$\exp \left(i\varphi \right)=e^{i\varphi }=\cos \left(\varphi \right)+i\sin \left(\varphi \right).$$

Using this relation, a complex sum of sine and cosine tin can exist expressed conveniently using a unmarried exponential function. This leads straight to the exponential note of complex numbers z = Ae ^iϕ. We will use the complex numbers and dissimilar notations in Sec. 2.3.

2.ii.ii. Convolution

As mentioned to a higher place, convolution is the operation that is necessary to describe the processing of any signal with a linear shift-invariant system. Convolution in the continuous case is divers equally

$$k(t)=(h*f)(t)={\displaystyle {\int }_{-\infty }^{\infty }h}(\tau )f(t-\tau )\text{d}\tau .$$

In society for the convolution to be well-divers, some requirements for the functions h and f must be fulfilled. For the space integral to be, h and f must decay fast enough towards infinity. This is the example if 1 of the functions has compact back up, i. e., it is 0 everywhere except for a express region. As an example, the convolution of a square input function f(t) with an Gaussian function h(t) is investigated in Geek Box ii.5. Farther mathematical properties of convolution are listed in Tabular array 2.ane.

Tabular array 2.one

Some mathematical properties of convolution, a, b are constants.

A mutual basic signal is the Dirac role which is as well called delta part or impulse function. It is a infinitely short, infinitely high impulse.

$$\delta (t)=\{\begin{array}{ll}\infty ,\hfill & \text{ if}t=0\hfill \\ 0,\hfill & \text{ otherwise}\hfill \end{array}$$

It is impossible to depict the Dirac function using classical functions. It requires the utilise of generalized functions or distributions, which is out of the scope of this introduction. The Dirac function is unremarkably represented graphically as an arrow of length one, see Fig. 2.2.

Figure two.2

Graphical representation of the Dirac function (t). The arrow symbolizes infinity.

Sequences of Dirac pulses are useful to select only certain points of a function like a sifter (cf. Figure 2.three). The sifting property of the Dirac function is given past integrating the product of a office and a time-delayed Dirac function

Geek Box 2.5 Convolution Example

For the definition of the foursquare function, the Heaviside pace part is useful to shorten the notation

$$H(t)=\{\begin{array}{ll}0,\hfill & \text{ if}t<0\hfill \\ 1,\hfill & \text{ otherwise}\hfill \end{array}.$$

And then, the square part and the Gaussian are defined equally

$$\begin{array}{l}f(t)=k_1+k_2{\displaystyle \sum _{\mathrm{due\; north}=-\infty }^{\infty }H}(t-nT)-H\left(t-\mathrm{north}T-k_3\right)\hfill \\ h(t)=\frac{1}{\sqrt{2\pi \sigma }}{\mathrm{due\; east}}^{-\frac{1}{\mathrm{two}}{\left(\frac{t}{\sigma }\right)}^{\mathrm{ii}}},\hfill \end{array}$$

with the starting time g ₁, the aamplitude thousand _ii, the duty-bicycle g _iii, and the period T of the square function and the standard difference σ of the Gaussian. The convolution with a Gaussian results in a smoothing of the edges of the square function.

With the sifting holding, the element at t = T can exist selected from the function, which is equivalent to sampling the role at that time point.

Figure 2.3

Laboratory sifters are used to remove undesired parts from discrete signals. Sequences of Dirac pulses can be applied in a like way. Image courtesy of BMK Wikimedia.

The sift property is useful for convolution of an capricious role and the Dirac part.

$$f(t)*\delta (t-T)={\displaystyle {\int }_{-\infty }^{\infty }f}(\tau )\delta (t-T-\tau )\text{d}\tau =f(t-T)$$

Consequently, the Dirac role is the identity element of convolution.

The response of a system to a Dirac office on the input is called the impulse response of the system $h(t)=\mathcal{H}\{\delta (t)\}$ . Using the superposition principle, every signal can be represented as a linear combination of infinitely many Dirac functions. Therefore, the output of a system to any input signal is computed by convolution of the input indicate f(t) with the impulse response h(t).

For medical applications, an important example of a linear shift-invariant system is an imaging system. The output of an imaging arrangement is often modeled as a linear shift-invariant organisation. The impulse response of an imaging organisation is chosen signal spread function. It describes how a single signal, i. eastward., a Dirac impulse, is spread on the sensor plane past the specific imaging system. The betoken spread function is a description of the behavior of the system.

two.ii.3. Correlation

Another basic functioning to combine a signal and a system is correlation

$$g(t)=(h\star f)(t)={\displaystyle {\int }_{-\infty }^{\infty }{h}^{*}}(\tau )f(t+\tau )\text{d}\tau ,$$

where h* is the complex cohabit of h. The main difference to convolution is that the input bespeak f is not mirrored earlier combination with h, i. e., f(t + τ) instead of f(t + τ). Correlation is a way to measure the similarity of ii signals.

An application of correlation is the matched filter. The matched filter is specifically designed to have a high response for a specific deterministic indicate or waveform f(t). It is matched to that point. The matched filter is direct computed by correlation with the desired signal. Alternatively, convolution with an impulse response of the mirrored, complex conjugate of the desired deterministic signal h(t) = f* (−t) can be used.

Technical uses for correlation can be establish in signal manual and signal detection. For a medical case, the heartbeats of a person can be detected in an Electrocardiogram (ECG) using correlation with a template QRS complex (QRS complex denotes the combination of three of the graphical deflections seen on an ECG). In image processing, a certain deterministic betoken is searched for across the whole prototype. In this instance, the deterministic betoken is ofttimes called template and the process of searching is called template matching. This tin can be used for the detection of specific structures and tracking of structures over time. Geek Box ii.6 puts the correlation in indicate processing in relation to the statistical correlation coefficient.

ii.3. Fourier Transform

Upwardly to this point, all operations and mathematical definitions were performed in continuous domain. Too, nosotros have not discussed the relation between detached and continuous representations which are important to understand the concept of sampling. In the following, we will innovate the Fourier transform and related concepts which volition permit us to deal with exactly such issues.

2.3.1. Types of Fourier Transforms

A cosine wave f of time t with amplitude A, frequency ξ, and phase shift φ tin be described by the following three equivalent parametrizations.

Geek Box ii.six Relation to the Statistical Correlation Coefficient

In statistics, the so-called Pearson correlation coefficient r [v] is a measure of agreement between two sets of observations x and y . Coefficient r is defined in the interval [−1, 1] and if |r| = ane, a perfect linear relationship between the ii variables is present. It is computed in the following mode:

$$r(x,y)=\frac{{\displaystyle \sum _{\mathrm{northward}}\left(x_n-\overline{x}\right)}\left(y_n-\overline{y}\right)}{{\sigma }_x{\sigma }_y}$$

Here, we use $\overline{\mathrm{10}}$ , ȳ, σ _ten , and σ _y to announce the corresponding mean values and standard deviations. If we assume the standard deviations to be equal to one and the means equal to 0, we go far at the following equation:

This is identical to the detached version of correlation for real inputs for t = 0. Also notation that this can exist considered but equally an inner product x ^⊤ y .

The prototype at the top of the page shows a scatter plot between two variables word recognition rate and expert rater. Each point (x _north , y _n) denotes one patient for whom both of the two variables were measured. The closer the 2 are to the dotted line, the better their agreement. Hither, their dependency is negative every bit if ane variable is loftier, the other is low and vice versa. r ≈ −0.9 in this example. Please refer to [four] for more details.

Figure 2.4

Approximation of a periodic signal using a weighted sum of trigonometric functions.

$$\begin{array}{cc}f(t)=A\cdot \cos (2\pi \xi t+\phi )& A,\phi \in ℝ\\ =a\cdot \cos (2\pi \xi t)+b\cdot \sin (\mathrm{two}\pi \xi t)& a,b\in ℝ\\ =c\cdot {\mathrm{eastward}}^{2\pi i\xi t}+\overline{c}\cdot {\mathrm{eastward}}^{-\mathrm{ii}\pi i\xi t}& c\in ℂ\end{array}$$

In Geek Box 2.7, nosotros show how the parameters a, b, and c are related to A and φ.

A Fourier serial (cf. Geek Box two.eight) is used to represent a continuous signal using only discrete frequencies. As such a Fourier series is able to estimate whatsoever signal equally a superposition of sine and cosine waves. Fig. 2.4(b) shows a rectangular point of time. The accented values of its Fourier coefficients are depicted in Fig. ii.four(a). As can be seen in Fig. 2.iv(a), the Fourier coefficients subtract every bit the frequency increases. It is therefore possible to approximate the signal by setting the coefficients to 0 for all high frequencies. Fig. two.iv(b) includes the approximations for 3 unlike choices of sets of frequencies.

The Fourier series, which works on periodic signals, can be extended to aperiodic signals by increasing the period length to infinity. The resulting transform is called continuous Fourier transform (or simply Fourier transform, cf. Geek Box 2.ix). Fig. two.five(b) shows the Fourier transform of a rectangular part, which is identical to the Fourier coefficients at the respective frequencies upwardly to scaling (see Fig. 2.5(a)).

The counter function to the Fourier series for cases in which fourth dimension domain is discrete and the frequency domain is continuous is called the discrete time Fourier transform (cf. Geek Box 2.ten). It forms a step towards the detached Fourier transform (cf. Geek Box 2.11) which allows us to perform all previous operations as well in a digital indicate processing organisation. In discrete space, nosotros tin can interpret the Fourier transform only as a matrix multiplication with a complex matrix F

where the betoken n and the discrete spectrum one thousand are vectors of complex values. The inverse operation is so readily found every bit

where F ^H is the Hermitian, i.e., transposed and element-wise conjugated, of F . Geek Box 2.12 shows some more details on how to observe these relations. Fig. ii.5 shows all types of Fourier transforms introduced in this section in comparison. Tab. 2.2 shows the Fourier transforms of popular functions.

Table 2.2

Fourier transforms of popular functions. Here nosotros apply the definition sinc(10)=sin(πx)πx. Notation that a convolution of 2 rectangular functions yields a triangular function as Ƒ[rect(t) * rect(t)] = sine²(ξ).

In computer programs, detached Fourier transforms are implemented very efficiently using fast Fourier transform (FFT). This approach reduces the number of computations from the society of N ⁱⁱ to the social club of Northward log N, if N is the length of the betoken. In the side by side department, we volition meet why convolution and correlation besides benefit from this efficiency.

ii.3.two. Convolution Theorem & Properties

The convolution of ii functions f and g is defined equally in Sec. ii.2.2, and denotes point-wise multiplication. The convolution theorem states that a convolution of two signals in space is identical to a point-wise multiplication of their spectra (see Equation 2.24). The contrary besides holds true (see Equation 2.25).

Geek Box 2.7 Equivalent Cosine Representations

Oscillations of the same frequency can be represented in several equivalent ways. In the following, we make use of the complex numbers introduced in Sec. two.2.i and the correspondence between a sum of complex exponentials and the real part $z+\overline{z}=2\mathrm{Re}(z)$ to convert the dissimilar representations into the aforementioned expression.

Aamplitude and phase shift, where we define $c=\frac{1}{2}A{\text{due east}}^{i\phi }$ :

$$\begin{array}{c}f(t)=A\cdot \cos (\mathrm{ii}\pi \xi t+\phi )=\mathrm{Re}\left(A\cdot e^{2\pi i\xi t+i\phi }\right)\\ =\mathrm{Re}\left(A\cdot e^{i\phi }\cdot {\mathrm{east}}^{2\pi i\xi t}\right)=\underset{\_}{\underset{\_}{\mathrm{Re}\left(2c\cdot {\mathrm{due\; east}}^{2\pi i\xi t}\right)}}.\end{array}$$

Sum of cosine and sine functions, where we define $c=\frac{\mathrm{i}}{\mathrm{ii}}(a-ib)$ :

$$\begin{array}{c}f(t)=a\cdot \cos (\mathrm{two}\pi \xi t)+b\cdot \sin (2\pi \xi t)\\ =a\cdot \cos (\mathrm{two}\pi \xi t)+b\cdot \cos (2\pi \xi t-\pi /2)\\ =\mathrm{Re}\left(a\cdot {\mathrm{east}}^{2\pi i\xi t}\right)+\mathrm{Re}\left(b\cdot {\mathrm{eastward}}^{\mathrm{ii}\pi i\xi t-\pi /2}\right)\\ =\mathrm{Re}\left(a\cdot e^{2\pi i\xi t}\right)+\mathrm{Re}\left(b\cdot e^{2\pi i\xi t}\cdot {\mathrm{eastward}}^{-i\pi /\mathrm{ii}}\right)\\ =\mathrm{Re}\left(a\cdot e^{2\pi i\xi t}\right)+\mathrm{Re}\left(-ib\cdot e^{2\pi i\xi t}\right)\\ =\mathrm{Re}\left((a-ib)\cdot {\mathrm{due\; east}}^{2\pi i\xi t}\right)=\underset{\_}{\underset{\_}{\mathrm{Re}\left(2c\cdot e^{2\pi i\xi t}\right).}}\end{array}$$

Sum of complex exponentials:

$$\begin{array}{c}f(t)=c\cdot {\text{e}}^{2\pi i\xi t}+\overline{c}\cdot {\text{e}}^{-\mathrm{two}\pi i\xi t}\\ =\mathrm{Re}\left(c\cdot {\text{eastward}}^{2\pi i\xi t}\right)+i\mathrm{Im}\left(c\cdot {\text{e}}^{\mathrm{two}\pi i\xi t}\right)+\mathrm{Re}\left(c\cdot {\text{due east}}^{\mathrm{ii}\pi i\xi t}\right)-i\mathrm{Im}\left(c\cdot {\text{e}}^{2\pi i\xi t}\right)\\ =\underset{\_}{\underset{\_}{\mathrm{Re}\left(\mathrm{two}c\cdot {\text{e}}^{2\pi i\xi t}\right).}}\end{array}$$

Geek Box 2.8 Fourier Serial

The Fourier serial (Equation 2.17) represents a periodic signal of period T past an space weighted sum of shifted cosine functions of different frequencies. The Fourier coefficients c are calculated using Equation ii.xvi.

$$c[k]=\frac{1}{T}{\displaystyle {\int }_d^{d+T}f\left(t\right)}{\text{e}}^{-2\pi itk/T}\text{dt}k\in \mathbb{Z}$$

$$f(t)={\displaystyle \sum _{\mathrm{thousand}=-\infty }^{\infty }c}[\mathrm{1000}]{\text{eastward}}^{2\pi itk/T}t\in ℝ$$

The coefficients c[k] and c[−k] together class a shifted cosine wave with frequency $\xi =\frac{\left|k\right|}{T}$ (see Geek Box two.vii). It follows that $c[-k]=\overline{c[k]}$ :

$$\begin{array}{c}c[k]{\text{east}}^{\mathrm{two}\pi itk/T}+c[-m]{\text{due east}}^{-2\pi itk/T}=c[k]{\text{e}}^{2\pi it\mathrm{one\; thousand}/T}+\overline{c[\mathrm{yard}]}{\text{eastward}}^{-2\pi it\mathrm{one\; thousand}/T}\\ c[-k]{\text{e}}^{-\mathrm{two}\pi itk/T}=\overline{c[k]}{\text{e}}^{-2\pi itk/T}\\ \Rightarrow c[-\mathrm{thousand}]=\overline{c[m]}\end{array}$$

Geek Box 2.nine Continuous Fourier Transform

Given a time-dependent signal f, its Fourier transform F at frequency ξ is defined past Eq. (2.18). The changed Fourier transform is defined by Eq. (2.19).

$$\begin{array}{cc}F(\xi )=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}f(t){\text{east}}^{-2\pi it\xi }\text{dt}& \xi \in ℝ\end{array}$$

In general, f(t) can be a circuitous signal. We will, all the same, only consider the example where f(t) is existent-valued. The continuous Fourier transform is symbolized by the operator $\mathcal{F}$ .

Geek Box 2.x Detached-time Fourier Transform

The spectrum (i.due east., continuous Fourier transform) of a band-express betoken that is sampled equidistantly and sufficiently dumbo with distance T can be calculated using the discrete-time Fourier transform (DTFT) divers by Equation two.20. The inverse transform is given by Equation 2.21. For details almost the required sampling distance encounter Sec. ii.four.ii.

$$\begin{array}{cc}{F}_{\frac{1}{T}}(\xi )={\displaystyle \sum _{\mathrm{north}=-\infty }^{\infty }f}[n]{\text{east}}^{-\mathrm{ii}\pi i\xi nT}& \xi \in ℝ\end{array}$$

$$\begin{array}{cc}f[\mathrm{northward}]=T{\displaystyle {\int }_d^{d+\frac{\mathrm{one}}{T}}{F}_{\frac{1}{T}}}(\xi ){\text{due east}}^{\mathrm{ii}\pi i\xi \mathrm{north}T}\text{d}\xi & \mathrm{due\; north}\in \mathbb{Z}\end{array}$$

Fig. 2.v(c) shows the DTFT of a band-express function and the Fourier transform. The DTFT is identical to the Fourier transform up to scaling except that information technology is periodic with flow 1/T.

Geek Box 2.eleven Discrete Fourier Transform

The spectrum of a periodic and band-express bespeak tin be calculated with the discrete Fourier transform (DFT) as defined by Equation 2.22. The signal can be reconstructed with the inverse DFT as defined by Equation 2.23.

$$\begin{array}{cc}F[\mathrm{chiliad}]={\displaystyle \sum _{\mathrm{north}=0}^{N-\mathrm{i}}f}[n]{\text{e}}^{-\mathrm{two}\pi i\mathrm{due\; north}k/\mathrm{North}}& k\in \mathbb{Z}\end{array}$$

$$\begin{array}{cc}f[n]=\frac{1}{\mathrm{Due\; north}}{\displaystyle \sum _{k=0}^{\mathrm{Northward}-\mathrm{i}}F}[k]{\text{east}}^{\mathrm{ii}\pi ink/\mathrm{Due\; north}}& n\in \mathbb{Z}\end{array}$$

Fig. 2.5(d) shows the DFT and the Fourier series of a band-limited signal. The DFT is identical to the Fourier series upward to scaling except that information technology is periodic with period 1/N.

Geek Box ii.12 Discrete Fourier Transform as Matrix

A discrete Fourier transform can be rewritten as a circuitous matrix product. To demonstrate this, we kickoff with the definition of the discrete Fourier transform:

$$\begin{array}{c}F[k]={\displaystyle \sum _{n=0}^{N-1}f}[n]{\text{e}}^{-2\pi in\mathrm{thou}/\mathrm{North}}\\ ={\displaystyle \sum _{n=0}^{\mathrm{Due\; north}-1}{\text{e}}^{-2\pi in\mathrm{grand}/N}}f[n]\end{array}$$

Now, we replace the summation with an inner production of two vectors ξ _k and n (cf. Geek Box ii.6):

$$F[\mathrm{1000}]=\left({\text{eastward}}^0,{\text{e}}^{-2\pi nk/\mathrm{North}},\dots ,{\text{due east}}^{-\mathrm{two}\pi i(\mathrm{North}-\mathrm{i})\mathrm{yard}/\mathrm{North}}\right)\left(\begin{array}{c}f[0]\\ f[1]\\ f[2]\\ ⋮\\ f[N-1]\end{array}\right)={\xi }_k^{\top }n$$

We see that ξ _yard is a discretely sampled wave at frequency k. This equation tin can now be interpreted as the k-th row of a matrix vector production. Thus, we can rewrite the entire discrete Fourier transform of all K frequencies to

$$\mathrm{thou}=\left(\begin{array}{c}F[0]\\ F[1]\\ ⋮\\ F[\mathrm{1000}-\mathrm{one}]\end{array}\right)=\left(\begin{array}{c}{\xi }_0^{\top }\\ {\xi }_{\mathrm{one}}^{\top }\\ ⋮\\ {\xi }_{K-\mathrm{i}}^{\top }\end{array}\right)n=Fn$$

As such, each row of the higher up matrix multiplication computes a correlation betwixt a wave of frequency k for all Thousand frequencies under consideration. Furthermore the relation F ^H = F ^−ane holds if F^H is scaled with $\frac{1}{\mathrm{North}}$ . Hence, F forms an orthonormal basis. If we keep this line of thought, we can as well interpret a Fourier transform as a basis rotation. In our case, we practice not rotate by a certain angle, only we projection our time-dependent signal into a frequency resolved time-contained space.

A similar theorem exists for the DFT. Allow C_h denote the matrix that performs the convolution with discrete impulse response h , and f be a discrete input signal. Then system output grand is obtained equally

where H is a diagonal matrix that contains the Fourier transformed coefficients of h . Note that F and F^H tin be implemented efficiently by means of FFT. In addition to the convolution theorem, the Fourier transform has other notable backdrop. Some of those properties are listed in Table 2.iii.

Table two.3

Furnishings of modifications of a signal in time on the Fourier transform.

two.4. Detached Organization Theory

two.4.1. Motivation

As already indicated in the introduction, discrete signals and systems are very important in practice. All signals tin can only be stored and processed at fixed detached time instances in a digital computer. The process of transforming a continuous time signal to a discrete time indicate is chosen sampling. In the simplest and near common case, the continuous signal is sampled at regular intervals, which is called uniform sampling. The electric current value of the continuous bespeak is stored exactly at the time instance where the discrete fourth dimension signal is defined. This can be modeled by a convolution with an impulse train, encounter Fig. 2.6(a). At first glance, it looks like a lot of information is discarded in the process of sampling. Withal, under certain requirements, the continuous time point can be reconstructed exactly. Further details are given in Sec. ii.4.ii.

As nosotros take already seen with the discrete Fourier transform, virtually methods introduced in this Chapter can exist every bit applied to detached signals. Nosotros denote discrete signals using brackets [] instead of parentheses (), as we already did in the Geek Boxes. Integrals must exist replaced past infinite sums, for case for the discrete convolution

$$\mathrm{1000}[n]=(h*f)[n]={\displaystyle \sum _{k=-\infty }^{\infty }h}[g]f[n-k].$$

In the detached example, the Dirac function takes on a simple grade.

$$\delta [n]=\{\begin{array}{ll}1,\hfill & \text{ if}n=0\hfill \\ 0,\hfill & \text{ otherwise}\hfill \end{array}$$

Note that in contrast to the continuous Dirac function, it is possible to exactly correspond and implement the discrete Dirac role.

In addition to the discrete contained variable, the dependent variable can also be discrete. This ways that the signal value f(t) or f[n] can only take values of certain levels. Apart from naturally discrete signals, all signals must be converted to a fixed detached value for representation and processing in digital computers. For instance, image intensities are often represented in the computer using 8 fleck, i.e., 256 different intensities, or 12 bit which corresponds to 4096 different levels. The process of transforming a continuous-valued signal to a discrete-valued signal is called quantization. In most cases, a uniform quantization is sufficient, which means that the detached levels have equal altitude from each other. The continuous-valued signal is rounded to the nearest discrete level available, see Fig. two.vi(b). The error arising during this process is called quantization noise. Some more than details on noise and noise models are given in Sec. 2.four.3.

2.iv.2. Sampling Theorem

The Nyquist-Shannon sampling theorem (or simply sampling theorem) states that a band-express signal, i.due east., a signal where all frequencies in a higher place ξ _B and below ξ _B are zero, can be fully reconstructed using samples 1/(2ξ _B ) autonomously. If we consider a sine moving ridge of frequency ξ _B , we have to sample it at least with a frequency of 2ξ _B , i.e. twice per wavelength.

Formally, the theorem can exist derived using the periodicity of the DTFT (come across Fig. two.5(c)). The DTFT spectrum is a periodic summation of the original spectrum, and the periodic spectra do not overlap as long every bit the sampling theorem is fulfilled. It is therefore possible to obtain the original spectrum past setting the DTFT spectrum to zilch for frequencies larger than B. The bespeak can then be reconstructed by applying the inverse Fourier transform. We refer to [iii] for a more than detailed clarification of this topic.

So far, we have not discussed how the actual sampling frequency 2ξ_B is determined. Luckily such a band limitation can be found for well-nigh applications. For example, even the most sensitive ears cannot perceive frequencies higher up 22 kHz. As a result, the sampling frequency of the compact disc (CD) was determined at 44.1 kHz. For the eye, typically 300 dots per inch in press or 300 pixels per inch for displays are considered as sufficient to prevent any visible distortions. In videos and films, a frame charge per unit of 50 Hz is ofttimes used to diminish flicker. Loftier fidelity devices may support up to 100 Hz.

If the sampling theorem is not respected, aliasing occurs. Frequencies above the Nyquist frequency are wrapped around due to the periodicity and announced every bit lower frequencies. Then, these high frequencies are indistinguishable from the truthful depression frequencies. Fig. ii.seven demonstrates this effect visually.

Figure two.7

Sampling a sine signal with a frequency below 2ξ _B will cause aliasing. The reconstructed sine moving ridge shown with blueish dashes does not lucifer the original frequency shown in ruddy.

2.4.3. Noise

In many cases, acquired measurements or images are corrupted by some unwanted signal components. Common racket sources are quantization and thermal noise. Boosted noise sources occur in the field of medical imaging, due to the related image acquisition techniques.

We can often detect a elementary model of the racket corrupting the image. The model does not represent the physical noise causes, simply it approximately describes the errors that occur in the final signal. An additive racket model is usually denoted as

where s(t) is the underlying desired indicate. We observe the indicate f(t), which is corrupted by the noise northward(t). For the statistics of the noise, we can use various models e.g., a Gaussian noise distribution $p(n(t))=\mathcal{N}\left(n(t)|{\mu }_n,{\Sigma }_n\right)$ . Another property of noise is its temporal or spatial correlation. This tin can be described past correlating the signal with itself, which is called autocorrelation part. An extreme case is white racket. White noise is temporally or spatially uncorrelated, pregnant the autocorrelation function is a Dirac impulse. The spectrum of white noise is constant, i.eastward., it contains all frequencies to the same amount as a white lite source would contain all visible wavelengths (cf. Fig. 2.8).

2.5. Examples

To conclude this chapter, we want to bear witness the introduced concepts of convolution and Fourier transform on two example systems. A elementary system is a smoothing filter, that allows only wearisome changes of the indicate. This is called a depression-pass filter. Information technology is an important building block in many applications, for example to remove high-frequency noise from a point or to remove point parts with loftier-frequency before downwards-sampling to avoid aliasing.

The filter coefficients of a low-pass filter are visualized in Fig. 2.9(a). The low-pass filter has a cutoff frequency of $\frac{\pi }{\mathrm{ii}}\frac{\text{ rad}}{\text{ sample}}$ and a length of 81 coefficients. The truthful properties of the low-pass filter are best perceived in the frequency domain, as displayed in Fig. 2.nine(b). Notation that the calibration of the y-axis is logarithmic. In this context, values of 0 betoken that the point can pass unaltered. Small-scale values indicate that the point components are damped. In this example, high frequencies are suppressed past several orders of magnitude. An ideal low-pass filter is a rectangle in the Fourier domain, i.eastward., all values beneath the cutoff frequency are passed unaltered and all values above are set to 0. In our discrete filter, we tin only estimate this shape. In the time-domain, the coefficients are samples of a sinc function, which is the changed Fourier transform of a rectangular function in Fourier domain (cf. Tab. 2.two). The opposite of the low-pass filter is the high-laissez passer filter, shown in Fig. ii.10. Hither, frequencies below the cutoff frequency are suppressed, whereas frequencies above are unaltered. Note that the time domain versions of loftier- and low-pass filters are difficult to differentiate.

Finally, we show how a signal with high and depression frequency components is transformed after convolution with a high-pass and a low-pass filter. The bespeak in Fig. ii.11 is a sine with condiment white racket. Thus, noise is distributed equally in the whole frequency domain. A large portion of the noise tin can exist removed by suppressing frequency components where no signal is present. Consequently, the cutoff frequency of the filters is slightly to a higher place the frequency of the sine function. Equally a result, the output of the loftier-pass filter is similar to the noise and the output of the low-pass filter is similar to the sine. In our example, we chose a causal filter which introduces a time delay in the filter output. A causal filter can only react to past inputs and needs to collect a certain corporeality of samples earlier the filtered upshot appears at the output.

Figure ii.11

Sine signal with condiment dissonance after processing with a depression-laissez passer filter and a high-laissez passer filter.

Further Reading

[1]

Ronald North Bracewell. The Fourier transform and its applications. McGraw-Loma, New York, 1986.

[2]

Olaf Dössel. Bildgebende Verfahren in der Medizin. Von der Technik zur medizinischen Anwendung. Vol. i. 2000.

[iii]

Bernd Girod, Rudolf Rabenstein, and Alexander Stenger. Einführung in dice Systemtheorie. Vol. iv. Teubner Stuttgart, 1997.

[4]

Andreas Maier. Speech communication of children with cleft lip and palate: Automatic assessment. Logos-Verlag, 2009.

[v]

Karl Pearson. "Mathematical contributions to the theory of evolution.—on a class of spurious correlation which may arise when indices are used in the measurement of organs". In: Proceedings of the regal order of london 60.359-367 (1897), pp. 489–498.

Source: https://www.ncbi.nlm.nih.gov/books/NBK546153/

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