A Continuous time System Has the Input output Relationship

Discrete Time System

Discrete-Time Signals and Systems

Luis F. Chaparro , Aydin Akan , in Signals and Systems Using MATLAB (Third Edition), 2019

Causality

In many situations signals need to be processed in real-time, i.e., the processing must be done as the signal comes into the system. In those situations, the system must be causal. Whenever the data can be stored, not a real-time situation, is not necessary to use a causal system.

A discrete-time system $\mathcal{S}$ is causal if:

•

whenever the input $x[n]=0$ , and there are no initial conditions, the output is $y[n]=0$ ,

•

the present output $y[n]$ does not depend on future inputs.

Causality is independent of the linearity and time-invariance properties of a system. For instance, the system represented by the input/output equation

$y[n]=x^2[n],$

where $x[n]$ is the input and $y[n]$ the output, is non-linear, time invariant, and according to the above definition causal: the output is zero whenever the input is zero, and the output depends on the present value of the input. Likewise, an LTI system can be noncausal, such is the case of the following LTI system that computes the moving average of the input:

$y[n]=\frac{1}{3}(x[n+1]+x[n]+x[n-1]).$

The input/output equation indicates that at the present time n to compute $y[n]$ we need a present value $x[n]$ , a past value $x[n-1]$ , and a future value $x[n+1]$ of the input. Thus, the system is LTI but noncausal since it requires future values of the input.

•

An LTI discrete-time system is causal if the impulse response of the system is such that

(9.37) $h[n]=0n<0.$

•

A signal $x[n]$ is said to be causal if

(9.38) $x[n]=0n<0.$

•

For a causal LTI discrete-time system with a causal input $x[n]$ its output $y[n]$ is given by

(9.39) $y[n]=\sum _{k=0}^{n}x[k]h[n-k]n\ge 0$

where the lower limit of the sum depends on the input causality, $x[k]=0$ for $k<0$ , and the upper limit on the causality of the system, $h[n-k]=0$ for $n-k<0$ or $k>n$ .

That $h[n]=0$ for $n<0$ is the condition for an LTI discrete-time system to be causal is understood by considering that when computing the impulse response, the input $\delta [n]$ only occurs at $n=0$ and there are no initial conditions so the response for $n<0$ should be zero. Extending the notion of causality to signals we can then see that the output of a causal LTI discrete-time system can be written in terms of the convolution sum as

$y[n]=\sum _{k=-\infty }^{\infty }x[k]h[n-k]=\sum _{k=0}^{\infty }x[k]h[n-k]=\sum _{k=0}^{n}x[k]h[n-k]$

where we first used the causality of the input ( $x[k]=0$ for $k<0$ ) and then that of the system, i.e., $h[n-k]=0$ whenever $n-k<0$ or $k>n$ . According to this equation the output depends on inputs $\{x[0],\cdots ,x[n]\}$ which are past and present values of the input.

Example 9.24

So far we have considered the convolution sum as a way of computing the output $y[n]$ of a LTI system with impulse response $h[n]$ for a given input $x[n]$ . But it actually can be used to find either of these three variables given the other two. The problem is then called deconvolution. Assume the input $x[n]$ and the output $y[n]$ of a causal LTI system are given, find equations to compute recursively the impulse response $h[n]$ of the system. Consider finding the impulse response $h[n]$ of a causal LTI system with input $x[n]=u[n]$ and output $y[n]=\delta [n]$ . Use the MATLAB function deconv to find $h[n]$ .

Solution: If the system is causal and LTI, the input $x[n]$ and the output $y[n]$ are connected by the convolution sum

$y[n]=\sum _{m=0}^{n}h[n-m]x[m]=h[n]x[0]+\sum _{m=1}^{n}h[n-m]x[m].$

To find $h[n]$ from given input and output values, under the condition that $x[0]\ne 0$ , the above equation can be rewritten as

$h[n]=\frac{1}{x[0]}[y[n]-\sum _{m=1}^{n}h[n-m]x[m]]$

so that the impulse response of the causal LTI can be found recursively as follows:

$\begin{array}{ccc}\hfill & \hfill \hfill & h[0]=\frac{1}{x[0]}y[0],h[1]=\frac{1}{x[0]}(y[1]-h[0]x[1]),\hfill \\ \hfill & \hfill \hfill & h[2]=\frac{1}{x[0]}(y[2]-h[0]x[2]-h[1]x[1])\cdots .\hfill \end{array}$

For the given case where $y[n]=\delta [n]$ and $x[n]=u[n]$ we get according to the above

$\begin{array}{ccc}\hfill & \hfill \hfill & h[0]=\frac{1}{x[0]}y[0]=1,\hfill \\ \hfill & \hfill \hfill & h[1]=\frac{1}{x[0]}(y[1]-h[0]x[1])=0-1=-1,\hfill \\ \hfill & \hfill \hfill & h[2]=\frac{1}{x[0]}(y[2]-h[0]x[2]-h[1]x[1])=0-1+1=0,\hfill \\ \hfill & \hfill \hfill & h[3]=\frac{1}{x[0]}(y[3]-h[0]x[3]-h[1]x[2]-h[2]x[3])=0-1+1-0=0,\cdots \hfill \end{array}$

and in general $h[n]=\delta [n]-\delta [n-1]$ .

The length of the convolution $y[n]$ is the sum of the lengths of the input $x[n]$ and of the impulse response $h[n]$ minus one. Thus,

$\text{length of}h[n]=\text{length of}y[n]-\text{length of}x[n]+1.$

When using the MATLAB function deconv we need to make sure that the length of $y[n]$ is always larger than that of $x[n]$ . If $x[n]$ is of infinite length, like when $x[n]=u[n]$ , this would require an even longer $y[n]$ , which is not possible. However, MATLAB can only provide a finite support input, so we make the support of $y[n]$ larger. In this example we have found analytically that the impulse response $h[n]$ is of length 2. Thus, if the length of $y[n]$ is chosen larger than the length of $x[n]$ by one we get the correct answer (case (a) in the script below). Otherwise we do not (case (b)). Run the two cases to verify this (get rid of the symbol % to run case (b)).

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Digital Signal Processing

Lars Wanhammar , in DSP Integrated Circuits, 1999

3.9 DIFFERENCE EQUATIONS

Many discrete-time and digital systems such as digital filters can be described by difference equations with constant coefficients. The input–output relation for an Nth-order LSI system can be described by

(3.11) $y\left(n\right)={\displaystyle \sum _{k=1}^N{b}_{k}y\left(n-k\right)+{\displaystyle \sum _{k=0}^M{a}_{k}x\left(n-k\right)}}$

It is customary to distinguish between LSI filters (systems) that have an impulse response of finite or infinite duration. The first type is called FIR (finite-length impulse response) filters since the impulse response becomes zero after a finite number of samples. The latter type is called IIR (infinite-length impulse response) filters since, even though the impulse response decays toward zero, it theoretically never reaches zero.

If b_k =   0 for all k, the system has a finite impulse response and is therefore an FIR system. Normally, the system is an IIR system if b_k   ≠   0 for at least one k, but there are some exceptions. If any b_k   ≠   0, the difference equation represents a recursive algorithm in which some of the previously computed output values are used to compute the next output value. It can be shown that IIR filters only can be realized by using recursive algorithms while FIR filters can be realized by using recursive or nonrecursive algorithms. However, it is not generally recommended that recursive algorithms be used to realize FIR filters, because of stability problems.

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Discrete-Time Signals and Systems

Luis F. Chaparro , in Signals and Systems using MATLAB, 2011

Causality

In many situations signals need to be processed in real time—that is, the processing must be done as the signal comes into the system. In those situations, the system must be causal. In many other situations, real-time processing is not required as the data can be stored and processed without the requirements of real time. Under such circumstances causality is not necessary.

A discrete-time system $\mathcal{S}$ is causal if:

■

Whenever the input x[n] = 0, and there are no initial conditions, the output is y[n] = 0.

■

The output y[n] does not depend on future inputs.

Causality is independent of the linearity and time-invariance properties of a system. For instance, the system represented by the input–output equation

$\begin{array}{l}\hfill y\left[n\right]=x^2\left[n\right]\end{array}$

where x[n] is the input and y[n] is the output is nonlinear but time invariant. According to the above definition it is a causal system: The output is zero whenever the input is zero, and the output depends on the present value of the input. Likewise, an LTI system can be noncausal, as can be seen in the following discrete-time system that computes the moving average of the input:

$\begin{array}{l}\hfill y\left[n\right]=\frac{1}{3}(x\left[n+1\right]+x\left[n\right]+x\left[n-1\right]).\end{array}$

The input–output equation indicates that at the present time n to compute y[n] we need a present value x[n], a past value x[n − 1], and a future value x[n + 1]. Thus, the system is LTI but noncausal since it requires future values of the input.

■

An LTI discrete-time system is causal if the impulse response of the system is such that

(8.36) $\begin{array}{l}\hfill h\left[n\right]=0n<0\end{array}$

■

A signal x[n] is said to be causal if

(8.37) $\begin{array}{l}\hfill x\left[n\right]=0n<0\end{array}$

■

For a causal LTI discrete-time system with a causal input x[n] its output y[n] is given by

(8.38) $\begin{array}{l}\hfill y\left[n\right]=\sum _{k=0}^{n}x\left[k\right]h\left[n-k\right]n\ge 0\end{array}$

where the lower limit of the sum depends on the input causality, x[k] = 0 for k < 0, and the upper limit depends on the causality of the system, h[n − k] = 0 for n − k < 0 or k > n.

That h[n] = 0 for n < 0 is the condition for an LTI discrete-time system to be causal is understood by considering that when computing the impulse response, the input δ[n] only occurs at n = 0 and there are no initial conditions, so the response for n < 0 should be zero. Extending the notion of causality to signals we can then see that the output of a causal LTI discrete-time system can be written in terms of the convolution sum as

$\begin{array}{l}\hfill y\left[n\right]=\sum _{k=-\mathfrak{\infty }}^{\mathfrak{\infty }}x\left[k\right]h\left[n-k\right]=\sum _{k=0}^{\mathfrak{\infty }}x\left[k\right]h\left[n-k\right]=\sum _{k=0}^{n}x\left[k\right]h\left[n-k\right]\end{array}$

where we first used the causality of the input (x[k] = 0 for k < 0) and then that of the system (i.e., h[n − k] = 0 whenever n − k < 0 or k > n). According to this equation the output depends on inputs {x[0], …, x[n]}, which are past and present values of the input.

Example 8.25

So far we have considered the convolution sum as a way of computing the output y[n] of an LTI system with impulse response h[n] for a given input x[n]. But it actually can be used to find either of these three variables given the other two. The problem is then called deconvolution. Assume the input x[n] and the output y[n] of a causal LTI system are given. Find equations to compute recursively the impulse response h[n] of the system. Consider finding the impulse response h[n] of a causal LTI system with input x[n] = u[n] and output y[n] = δ[n]. Use the MATLAB function deconv to find h[n].

Solution

If the system is causal and LTI, the input x[n] and the output y[n] are connected by the convolution sum

$\begin{array}{l}\hfill y\left[n\right]=\sum _{m=0}^{n}h\left[n-m\right]x\left[m\right]=h\left[n\right]x\left[0\right]+\sum _{m=1}^{n}h\left[n-m\right]x\left[m\right]\end{array}$

To find h[n] from given x[n] and y[n], under the condition that x[0] ≠ 0, the above equation can be rewritten as

$\begin{array}{l}\hfill h\left[n\right]=\frac{1}{x\left[0\right]}\left[y\left[n\right]-\sum _{m=1}^{n}h\left[n-m\right]x\left[m\right]\right]\end{array}$

so that the impulse response of the causal LTI can be found recursively as follows:

$\begin{array}{llll}\hfill h\left[0\right]& =\frac{1}{x\left[0\right]}y\left[0\right]& \hfill & \\ \hfill h\left[1\right]& =\frac{1}{x\left[0\right]}\left(y\left[1\right]-h\left[0\right]x\left[1\right]\right)& \hfill & \\ \hfill h\left[2\right]& =\frac{1}{x\left[0\right]}\left(y\left[2\right]-h\left[0\right]x\left[2\right]-h\left[1\right]x\left[1\right]\right)& \hfill & \\ \hfill & ⋮& \hfill & \end{array}$

For the given case where y[n] = δ[n] and x[n] = u[n], we get, according to the above,

$\begin{array}{llll}\hfill h\left[0\right]& =\frac{1}{x\left[0\right]}y\left[0\right]=1& \hfill & \\ \hfill h\left[1\right]& =\frac{1}{x\left[0\right]}\left(y\left[1\right]-h\left[0\right]x\left[1\right]\right)=0-1=-1& \hfill & \\ \hfill h\left[2\right]& =\frac{1}{x\left[0\right]}\left(y\left[2\right]-h\left[0\right]x\left[2\right]-h\left[1\right]x\left[1\right]\right)=0-1+1=0& \hfill & \\ \hfill h\left[3\right]& =\frac{1}{x\left[0\right]}\left(y\left[3\right]-h\left[0\right]x\left[3\right]-h\left[1\right]x\left[2\right]-h\left[2\right]x\left[3\right]\right)=0-1+1-0=0& \hfill & \\ \hfill & ⋮& \hfill & \end{array}$

and, in general, h[n] = δ[n] − δ[n − 1].

The length of the convolution y[n] is the sum of the lengths of the input x[n] and of the impulse response h[n] minus one. Thus,

$\begin{array}{l}\hfill \text{length}\text{of}h\left[n\right]=\text{length}\text{of}y\left[n\right]-\text{length}\text{of}x\left[n\right]+1\end{array}$

When using deconv we need to make sure that the length of y[n] is always larger than that of x[n]. If x[n] is of infinite length, like when x[n] = u[n], this would require an even longer y[n], which is not possible. However, MATLAB can only provide a finite-support input, so we make the support of y[n] larger. In this example we have found analytically that the impulse response h[n] is of length 2, so if the length of y[n] is chosen so that length y[n] is larger than the length of x[n] by one, we get the correct answer (case (a) in the script below); otherwise we do not (case (b)). Run the two cases to verify this (get rid of % symbol to run case (b)).

%%%%%%%%%%%%%%%%%%%%%%

% Example 8.25 --- Deconvolution

%%%%%%%%%%%%%%%%%%%%%%

clear all

x = ones(1, 100);

y = [1 zeros(1, 100)]; % (a) correct h

% y = [1 zeros(1, 99)]; % (b) incorrect h

[h, r] = deconv(y, x)

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Detection, Classification, and Estimation in the (t,f) Domain

In Time-Frequency Signal Analysis and Processing (Second Edition), 2016

12.3.1 Problem Description

A discrete time system identification problem can be stated as follows:

(12.3.1) $\begin{array}{l}\hfill y[n]=\sum _{k}q[n-k]x[k]+ϵ[n],\end{array}$

where x[n] is a transmitted signal, q[n] is the impulse response of a linear time invariant (LTI) system, ϵ[n] is an additive noise, and y[n] is the received signal. The problem is to identify the LTI system transfer function Q(f), i.e., the Fourier transform, of q[n] given the input and the output signals x[n] and y[n].

The conventional method for solving the above problem is the least-squares solution method that is equal to the cross-spectral method in stationary cases, i.e., the system transfer function Q(f) can be estimated by (see, e.g., Refs. [24,25])

(12.3.2) $\begin{array}{l}\hfill Q(f)=\frac{S_{xy}(f)}{S_{xx}(f)},\end{array}$

where S _xy (f) is the cross-spectrum of x[n] and y[n], and S _xx (f) is the auto-spectrum of x[n]. When the additive noise ϵ[n] in Eq. (12.3.1) is a zero-mean Gaussian process and statistically independent of the input signal x[n], the estimate in Eq. (12.3.2) is asymptotically unbiased but the performance is limited by the noise variance or the signal-to-noise ratio (SNR). When the SNR is low, the performance of the estimate in Eq. (12.3.2) is poor as we will see later. Since the auto-spectrum of the input signal x[n] is in the denominator in the estimate equation (12.3.2), the input signal is, in general, chosen as a pseudorandom signal with flat spectrum. With these types of input signals, noise reduction techniques before system identification do not apply.

In the following, we introduce a different technique [26] for the system identification problem. The main idea is as follows. Instead of pseudorandom signal x[n], chirp type signals are transmitted as training signals, which have wideband characteristics in the frequency domain but are concentrated in the joint time-frequency ((t,f)) domain. The (t,f) concentration property usually holds after passing through an LTI system (this will be seen later). Since a joint (t,f) distribution usually spreads noise and localizes signals, in particular chirps, the receiver may use a (t,f) analysis technique to map the received signal y[n] from the time domain into the joint (t,f) domain. In this way, the SNR can be significantly increased in the joint (t,f) domain [27]. Furthermore, (t,f) filtering can be used in the (t,f) plane to reduce the noise and the SNR in the time domain can be increased and therefore the system identification after denoising can be improved. Some applications of this approach can be found in Refs. [28–31].

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Signals and Systems

Rashid Ansari , Lucia Valbonesi , in The Electrical Engineering Handbook, 2005

Causality

If a discrete-time system is causal, then the current output sample depends only on the current and past input samples and not on future samples. This definition is now examined in the case of an LSI system with an impulse response h[n].

The output y[n] of an LSI system can be expressed as:

(1.30) $\begin{array}{ll}y[n]& =h[n]*x[n]=\underset{k=-\infty }{\overset{\infty }{\Sigma }}x[n-k]h[k]\\ & =\cdots +x[n+2]h[-2]+x[n+1]h[-1]\\ & +x[n]h[0]+x[n-1]h[1]+x[n-2]h[2]+\dots .\end{array}$

The expression of equation 1.30 immediately suggests the constraint on the impulse response of a causal LSI system. If the output is required to be independent of future samples of an arbitrary input, then the coefficients h[− 1], h[− 2], … in the convolution sum should each be equal to zero.

Therefore, an LSI system is causal if and only if h[n] = 0, n < 0.

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Discrete-Time Control System Implementation Techniques

Wei-Yong Yan , Robert R. Bitmead , in Control and Dynamic Systems, 1995

IV Conclusions

A PTV discrete-time system can be represented by an LTI system in a certain equivalent sense. Likewise, a PTV compensator can be represented by an LTI compensator. We have shown that the monodromy matrix of the closed-loop system consisting of the PTV system and compensator coincides with the state matrix of the closed-loop system consisting of the representing LTI system and compensator. However, this property does not imply that controlling the periodic system by periodic controllers can be completely reduced to controlling the equivalent LTI system by LTI controllers since not all LTI compensators represent some periodic compensator for the original periodic system. One of the features about the representing LTI system is the existence of direct feedthrough from input to output whether the original PTV system has direct control feedthrough or not.

Although it is often difficult to know whether an LTI compensator for the representing LTI system corresponds to a periodic compensator for the periodic system, it is clear that the periodic output feedback is equivalent to the constant static output feedback with a block diagonal gain matrix. Based on this equivalence, we have further identified the PTV system as an LTI multi-channel system so that the problem of controlling the PTV system by periodic output feedback can be reposed as that of controlling the LTI multi-channel system with direct control feedthrough by static decentralized output feedback. In addition, the concepts of periodic fixed modes and strong connectedness have been naturally introduced for PTV systems.

Decentralized control techniques have been discussed for general LTI multi-channel systems and applied to the problem of controlling PTV systems by periodic output feedback. As a result, whether a strongly connected periodic system is stabilizable or almost pole-assignable by periodic output feedback can be easily verified in terms of periodic fixed modes in the case where the number of states is less than the sum of the numbers of inputs, outputs, and periodic fixed modes. The procedure for designing a desired periodic output feedback is straightforward and explicit.

One obvious question which may be worth further research is as to whether periodic fixed modes can be removed by general PTV controllers. Also, it has not yet been resolved how to design PTV dynamic controllers to stabilize a given PTV system.

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REVIEWS

Wing-Kuen Ling , in Nonlinear Digital Filters, 2007

BACKGROUNDS ON BIFURCATION THEOREM

Suppose a discrete time system can be represented as x(k + 1) = Ax(k) + Bg(Cx(k); μ) and y(k) = Cx(k), where x(k) ∈ ℜ ⁿ , A ∈ ℜ ^nxn , y(k) ∈ℜ^m, g(·):ℜ^m × ℜ → ℜ^l is a smooth (Cr, r ≥ 3) l dimensional vector field, k ∈ Z ⁺∪ {0} is the iteration index, and μ∈ℜ is the bifurcation parameter. It is worth noting that all of the matrices may depend on μ, and A may be a zero matrix. By introducing an arbitrary matrix D∈ℜ ^lxm , which may also depend on μ, the system can be represented by a feedback system. The whole feedback system consists of the linear feedforward part G(z; μ) and the nonlinear feedback part f(e(k); μ) as shown in Figure 2.7, and the system can be described as x(k + 1) = Ax(k) + BDy(k) + B(g(Cx(k); μ) – Dy(k))or x(k + 1) = (A + BDC)x(k) + B(g(Cx(k); μ) – Dy(k)), G(z; μ) = C(z I – (A + BDC))⁻¹ B, u(k) = f(e(k); μ) + v(k) = g(y(k); μ) – Dy(k) and e(k) = d(k) – y(k). If we assume that the reference input v(k) = 0 and the external disturbance d(k) = 0, supposing that the system achieves equilibrium at $\text{e}(k)=\hat{\text{e}}$ , by linearizing the nonlinear feedback system f(e(k); μ) at the equilibrium point, then the open loop gain of the system is

Figure 2.7. Nonlinear feedback system.

$\begin{array}{ll}\text{G}(z;\mu )\text{J}(\mu ),\hfill & {\text{where\hspace{0.5em}J}(\mu )\equiv \frac{\partial \text{f}(\text{e}(k))}{\partial \text{e}(k)}|}_{\text{e}(k)=\hat{\text{e}}}\hfill \end{array}$

is the Jacobian matrix. If there exists a simple pair of complex eigenvalues of G(z; μ)J(μ) for z = e ^jω crossing the critical point −1 + 0j for some values of ω₀ and a given parameter μ = μ₀, then the system exhibits the Hopf bifurcation.

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Discrete-time Signals and Systems

Luis Chaparro , in Signals and Systems Using MATLAB (Second Edition), 2015

9.3.2 Dynamic Discrete-time Systems Represented by Difference Equations

A recursive discrete-time system is represented by a difference equation

(9.30) $\begin{array}{cc}\hfill & y[n]=-{\displaystyle \sum _{k=1}^{N-1}}{a}_{k}y[n-k]+{\displaystyle \sum _{m=0}^{M-1}}{b}_{m}x[n-m]n\ge 0\hfill \\ \hfill & \text{initial}\text{conditions}y[-k]\text{,}k=1\text{,}\dots \text{,}N-1\hfill \end{array}$

which naturally characterizes the dynamics of the system. On the other hand, this difference equation could be the approximation of an ordinary differential equation representing a continuous-time system being processed discretely. For instance, to approximate a second-order ordinary differential equation by a difference equation, we could approximate the first derivative as

$\frac{{\mathit{dv}}_c(t)}{\mathit{dt}}\approx \frac{v_c(t)-v_c(t-T_s)}{T_s}$

and the second derivative as

$\frac{d^2{v}_c(t)}{{\mathit{dt}}^2}=\frac{d\frac{{\mathit{dv}}_c(t)}{\mathit{dt}}}{\mathit{dt}}\approx \frac{d(v_c(t)-v_c(t-T_s))/T_s}{\mathit{dt}}=\frac{v_c(t)-2v_c(t-T_s)+v_c(t-2T_s)}{T_s^2}$

to obtain a second-order difference equation. Choosing a small value for T _s provides an accurate approximation to the ordinary differential equation. Other transformations can be used. In Chapter 0 we indicated that approximating integrals by the trapezoidal rule gives the bilinear transformation which can also be used to change differential into difference equations.

Just as in the continuous-time case, the system being represented by the difference equation is not LTI unless the initial conditions are zero and the input is causal. The Z-transform will however allow us to find the complete response of the system even when the initial conditions are not zero.

The complete response of a system represented by the difference equation can be shown to be composed of a zero-input and a zero-state responses, i.e., if y[n] is the solution of the difference Equation (9.30) with initial conditions not necessarily equal to zero, then

(9.31) $y[n]=y_{\mathit{zi}}[n]+y_{\mathit{zs}}[n]\text{.}$

The component y _zi [n] is the response when the input x[n] is set to zero, thus it is completely due to the initial conditions. The response y _zs [n] is due to the input only, as we set the initial conditions to zero. The complete response y[n] is thus seen as the superposition of these two responses. The Z-transform provides an algebraic way to obtain the complete response, whether the initial conditions are zero or not. It is important, as in the continuous-time, to differentiate the zero-input and the zero-state responses from the transient and the steady-state responses. Examples illustrating how to obtain these responses using the Z-transform will be given in the next chapter.

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Finite impulse response filter design

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

6.1 Classification of digital filters

Digital filters are discrete-time systems. The type of digital filters that we shall design in this course is linear. Therefore, they possess all the properties of linear discrete-time systems discussed in Chapter 3. All linear discrete-time operations on an input sequence can be viewed as a filtering of the sequence to produce an output sequence. This is the reason why digital filters are so important in DSP.

Non-linear filters are also commonly used, especially in areas such as image processing. The median filter discussed in section 5.4 for image enhancement is a typical non-linear digital filter.

Linear systems are characterized by their impulse responses. An impulse response can either have a finite or an infinite duration. A finite impulse response h(n) has its non-zero values extending over a finite time interval and is zero beyond that interval. The following finite impulse response

$h\left(n\right)=\left\{h_0,h_1,h_2,\dots ,h_N,0,0,0,\dots \right\}$

has non-zero values in the interval

and is referred to as a finite impulse response (FIR) filter or system of order N. So an Nth order FIR digital filter has an impulse response with a length of (N+1) samples. The samples of the impulse response function (h ₀, h ₁, etc) are usually called filter coefficients, filter weights, and filter tap coefficients/weights.

If the impulse response function has an infinite duration, we have an infinite impulse response (IIR) filter. It is obvious that IIR filters cause computational problems since we cannot compute an infinite number of terms. But the type of IIR filters that are designed have their input and output samples interrelated through a linear difference equation. The output sequence can then be computed recursively. This is the reason why IIR filters are also known as recursive filters and FIR filters as non-recursive filters.

In this chapter, we shall concentrate on FIR filters. IIR filters will be discussed in detail in the next chapter.

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Kalman Filters and Nonlinear Filters

Vytas B. Gylys , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.C Nonlinear System Model

We now generalize the discrete-time system model defined in terms of Eqs. (2) and (3) to

(29) ${\text{s}}_{\text{k}}={\text{f}}_{\text{k}-1}\left({\text{s}}_{\text{k}-1},{\text{u}}_{\text{k}-1}\right)+{\text{G}}_{\text{k}-1}\left({\text{s}}_{\text{k}-1}\right){\text{w}}_{\text{k}-1}$

and

(30) ${\text{m}}_{\text{k}}={\text{h}}_{\text{k}}\left({\text{s}}_{\text{k}}\right)+{\text{v}}_{\text{k}}.$

We do it by replacing the linear terms Φ _{k  −   1} s _{k  −   1} and B _{k  −   1} u _{k  −   1} of the linear dynamics model by a function f _k (·,·), allowing the process noise-to-state transformation matrix G _{k  −   1} to depend on system state, and generalizing the state-to-measurement transformation matrix H _k to a function h _k (·). The subscripts attached to f, G, and h emphasize that these functions may be of time-dependent form. Furthermore, f may be nonlinear in s and possibly in u; h may be nonlinear in s. The same distributional assumptions as made for the linear discrete-time model (Assumptions 1D–4D) are used for the nonlinear model defined by Eqs. (29) and (30). Now as before, the noise is included in the model through additive terms; also, functions f and h in general are vector-valued so that their dimensions, respectively, agree with the lengths of vectors s and m. More general nonlinear system models are possible.

One could similarly define a continuous-time nonlinear system model, but we do not do it in this chapter.

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