Discrete time convolution

1.1 Units. Throughout this semester, we will use the integer-valued variable n as the time variable for discrete-time signal processing; that is, ...

The identity under convolution is the unit impulse. (t0) gives x 0. u (t) gives R t 1 x dt. Exercises Prove these. Of the three, the first is the most difficult, and the second the easiest. 4 Time Invariance, Causality, and BIBO Stability Revisited Now that we have the convolution operation, we can recast the test for time invariance in a new ...Operation Definition. Continuous time convolution is an operation on two continuous time signals defined by the integral. (f ∗ g)(t) = ∫∞ −∞ f(τ)g(t − τ)dτ ( f ∗ g) ( t) = ∫ − ∞ ∞ f ( τ) g ( t − τ) d τ. for all signals f f, g g defined on R R. It is important to note that the operation of convolution is commutative ...Convolutions De nition/properties Convolution theorem Transfer function, Laplace vs. time space solutions 1 Introduction (what is the goal?) A car traveling on a road is, in its simplest form, a mass on a set of springs (the shocks). Bumps on the road apply a force that perturbs the car. A (very) simple model might takeDiscrete-Time Convolution Example: "Sliding Tape View" D-T Convolution Examples x n [ n ] = ( 1 ) 2 u [ n ] [ n ] = u [ n ] − u [ n − 4 ] h [i ] x [i ] ... i -3 -2 -1 1 2 3 4 5 6 7 8 9 Choose to flip and slide h[n] [ 0 − i ] This shows h[n-i] for = 0 For n < 0 h[n-i]x(i) = 0 ∀i ⇒ y [ n ] = 0 forThe proof of the frequency shift property is very similar to that of the time shift (Section 9.4); however, here we would use the inverse Fourier transform in place of the Fourier transform. Since we went through the steps in the previous, time-shift proof, below we will just show the initial and final step to this proof: z(t) = 1 2π ∫∞ ...http://adampanagos.orgThis video works an example of discrete-time convolution using the "reflect, shift, and sum" approach. Basically, this means we sketch...EEL3135: Discrete-Time Signals and Systems Discrete-Time Systems, LTI Systems, and Discrete-Time Convolution - 3 - (10) Note that we simply replaced with in equation (9) to produce . Next, we follow the bot-tom path in the diagram: (11) Note that in this case, we first compute [equation (9)] and then replace with . Since (10) and

where x*h represents the convolution of x and h. PART II: Using the convolution sum The convolution summation is the way we represent the convolution operation for sampled signals. If x(n) is the input, y(n) is the output, and h(n) is the unit impulse response of the system, then discrete- time convolution is shown by the following summation.The Discrete-Time Convolution (DTC) is one of the most important operations in a discrete-time signal analysis [6]. The operation relates the output sequence y(n) of a linear-time invariant (LTI) system, with the input sequence x(n) and the unit sample sequence h(n), as shown in Fig. 1.Figure 1 shows an example of such a convolution operation performed over two discrete time signals x 1 [n] = {2, 0, -1, 2} and x 2 [n] = {-1, 0, 1}. Here the first and the second rows correspond to the original signal x 1 [n] and flipped version of the signal x 2 [n], respectively. Figure 1. Graphical method of finding convolutionDigital Signal Processing Questions and Answers – Analysis of Discrete time LTI Systems ... Convolution sum b) Convolution product c) Convolution Difference d) None of the mentioned View Answer. Answer: a Explanation: The input x(n) is convoluted with the impulse response h(n) to yield the output y(n). As we are summing the different values ...Discretion is a police officer’s option to use his judgment to interpret the law as it applies to misdemeanor crimes. The laws that apply to felony crimes, such as murder, are black and white.Discrete-Time-Convolution LTI Systems. A system which produces an output signal from any input signal subject to constraints linearity and time invarience. Such a system is called Linear Time Invariant(LTI) System . Let's say x[n] is an input signal and y[n] is the output signal of the system.More seriously, signals are functions of time (continuous-time signals) or sequences in time (discrete-time signals) that presumably represent quantities of interest. Systems are operators that accept a given signal (the input signal) and produce a new signal (the output signal). Of course, this is an abstraction of the processing of a signal.

This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Continuous Time Convolution – 2”. For all the following problems, h*x denotes h convolved with x. $ indicates integral. 1. Find the value of [d (t) – d (t-1)] * -x [t+1]. a) x (t+1) – x (t) b) x (t) – x (t+1) c) x (t) – x (t-1) d) x (t-1) – x ...The behavior of a linear, time-invariant discrete-time system with input signal x [n] and output signal y [n] is described by the convolution sum. The signal h [n], assumed known, is the response of the system to a unit-pulse input. The convolution summation has a simple graphical interpretation.The Discrete Fourier Transform (DFT) Midterm Exam 16 Linear Filtering with the DFT 17 Spectral ... FFT Algorithms 20 The Goertzel Algorithm and the Chirp Transform 21 Short-time Fourier Analysis 22 Modulated Filter Bank 23 Caruso’s Orchestra Final Exam Course Info Instructor Prof. Alan V. Oppenheim; Departments Electrical Engineering and ...Example #3. Let us see an example for convolution; 1st, we take an x1 is equal to the 5 2 3 4 1 6 2 1. It is an input signal. Then we take impulse response in h1, h1 equals to 2 4 -1 3, then we perform a convolution using a conv function, we take conv(x1, h1, ‘same’), it performs convolution of x1 and h1 signal and stored it in the y1 and y1 has a length of 7 because we use a shape as a same.d) x [n] + h [n] View Answer. 3. What are the tools used in a graphical method of finding convolution of discrete time signals? a) Plotting, shifting, folding, multiplication, and addition in order. b) Scaling, shifting, multiplication, and addition in order. c) Scaling, multiplication and addition in order. Discrete data refers to specific and distinct values, while continuous data are values within a bounded or boundless interval. Discrete data and continuous data are the two types of numerical data used in the field of statistics.

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(ii) Ability to recognize the discrete-time system properties, namely, memorylessness, stability, causality, linearity and time-invariance (iii) Understanding discrete-time convolution and ability to perform its computation (iv) Understanding the relationship between difference equations and discrete-time signals and systemsIt lets the user visualize and calculate how the convolution of two functions is determined - this is ofen refered to as graphical convoluiton. The tool consists of three graphs. Top graph: Two functions, h (t) (dashed red line) and f (t) (solid blue line) are plotted in the topmost graph. As you choose new functions, these graphs will be updated.This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Concept of Convolution”. 1. The resulting signal when a continuous time periodic signal x (t) having period T, is convolved with itself is ___________. a) Non-Periodic. b) Periodic having period 2T. c) Periodic having period T. d) Periodic having period T/2.Discrete Time Convolution Example. 1. Discrete Time Fourier Analysis. 3. Contradiction while using the convolution sum for a non-LTI system. 3. Representing a continuous LTI system as a discrete one. 3. LTI, causal, discrete time system output. 2. Convolution of 2 discrete time signals. 3.

Discrete convolution tabular method. In the time discrete convolution the order of convolution of 2 signals doesnt matter : x1(n) ∗x2(n) = x2(n) ∗x1(n) x 1 ( n) ∗ x 2 ( n) = x 2 ( n) ∗ x 1 ( n) When we use the tabular method does it matter which signal we put in the x axis (which signal's points we write 1 by 1 in the x axis) and which ...May 22, 2022 · Discrete Time Fourier Series. Here is the common form of the DTFS with the above note taken into account: f[n] = N − 1 ∑ k = 0ckej2π Nkn. ck = 1 NN − 1 ∑ n = 0f[n]e − (j2π Nkn) This is what the fft command in MATLAB does. This modules derives the Discrete-Time Fourier Series (DTFS), which is a fourier series type expansion for ... 25-Apr-2023 ... The convolution operator is frequently used in signal processing to simulate the impact of a linear time-invariant system on a signal. In ...1.1 Units. Throughout this semester, we will use the integer-valued variable n as the time variable for discrete-time signal processing; that is, ...Operation Definition. Continuous time convolution is an operation on two continuous time signals defined by the integral. (f ∗ g)(t) = ∫∞ −∞ f(τ)g(t − τ)dτ ( f ∗ g) ( t) = ∫ − ∞ ∞ f ( τ) g ( t − τ) d τ. for all signals f f, g g defined on R R. It is important to note that the operation of convolution is commutative ...1 Answer. Sorted by: 1. The multiplication of the two unit step sequences u[k] ⋅ u[−n + k − 1] u [ k] ⋅ u [ − n + k − 1] is only non-zero if both sequences are non-zero. This means that the condition k ≥ 0 k ≥ 0 as well as the condition k ≥ n + 1 k ≥ n + 1 must be satisfied. So you have two cases: for n <= −1 n <= − 1 ...Convolution Theorem. Let and be arbitrary functions of time with Fourier transforms . Take. (1) (2) where denotes the inverse Fourier transform (where the transform pair is defined to have constants and ). Then the convolution is.Operation Definition. Discrete time convolution is an operation on two discrete time signals defined by the integral. (f ∗ g)[n] = ∑k=−∞∞ f[k]g[n − k] for all signals f, g defined on Z. It is important to note that the operation of convolution is commutative, meaning that. f ∗ g = g ∗ f. for all signals f, g defined on Z.

Discrete-Time-Convolution LTI Systems. A system which produces an output signal from any input signal subject to constraints linearity and time invarience. Such a system is called Linear Time Invariant(LTI) System . Let's say x[n] is an input signal and y[n] is the output signal of the system.

Discrete-Time Convolution Array. x[N] . h[M] . x[N]h[M] . y[N+M] x[N+1] . h[M+1] . …The transfer function is a basic Z-domain representation of a digital filter, expressing the filter as a ratio of two polynomials. It is the principal discrete-time model for this toolbox. The transfer function model description for the Z-transform of a digital filter's difference equation is. Y ( z) = b ( 1) + b ( 2) z − 1 + … + b ( n + 1 ...Dividends are corporate profits paid out to company stockholders. Dividends are declared by the board of directors and are typically paid quarterly, but there are several exceptions in which dividends can be paid more or less often. Dividen...w = conv (u,v) returns the convolution of vectors u and v. If u and v are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials. w = conv (u,v,shape) returns a subsection of the convolution, as specified by shape . For example, conv (u,v,'same') returns only the central part of the convolution, the ... 5.1 The discrete-time Fourier transform. As we have seen in the previous chapter, the complex exponential is an eigenfunction of LTI systems. That is, if the input \(e^{j\omega_0 n}\) is given to an LTI system, the output is just a scaled version of the same.The identity under convolution is the unit impulse. (t0) gives x 0. u (t) gives R t 1 x dt. Exercises Prove these. Of the three, the first is the most difficult, and the second the easiest. 4 Time Invariance, Causality, and BIBO Stability Revisited Now that we have the convolution operation, we can recast the test for time invariance in a new ...Therefore, a discrete time sliding mode predictive control for overhead …10.1: Signal Sampling. This module introduces sampling of a continuous time signal to produce a discrete time signal, including a computation of the spectrum of the sampled signal and a discussion of its implications for reconstruction. 10.2: Sampling Theorem. This module builds on the intuition developed in the sampling module to discuss the ...So the impulse response of filters arranged in a series is a convolution of their impulse responses (Figure 3). Figure 3. Associativity of the convolution enables us to exchange successive filters with a single filter whose impulse response is a convolution of the initial filters’ impulse responses. Proof for the discrete caseSteps for Graphical Convolution. First of all re-write the signals as functions of τ: x(τ) and h(τ) Flip one of the signals around t = 0 to get either x(-τ) or h(-τ) Best practice is to flip the signal with shorter interval. We will flip h(τ) to get h(-τ) throughout the steps. Determine Edges of the flipped signal.

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ECE 314 – Signals and Communications Fall/2004 Solutions to Homework 5 Problem 2.33 Evaluate the following discrete-time convolution sums: (a) y[n] = u[n+3]∗u[n−3] Lecture 1 : Introduction. Objectives. In this lecture you will learn the following. First of all we will try to look into the formal definitions of the terms ' signals ' and ' systems ' and then go on further to introduce to you some simple examples which may be better understood when seen from a signals and systems perspective.Periodic convolution is valid for discrete Fourier transform. To calculate periodic convolution all the samples must be real. Periodic or circular convolution is also called as fast convolution. If two sequences of length m, n respectively are convoluted using circular convolution then resulting sequence having max [m,n] samples.The Discrete-Time Convolution (DTC) is one of the most important operations in a discrete-time signal analysis [6]. The operation relates the output sequence y(n) of a linear-time invariant (LTI) system, with the input sequence x(n) and the unit sample sequence h(n), as shown in Fig. 1 . The Discrete-Time Convolution (DTC) is one of the most important operations in a discrete-time signal analysis [6]. The operation relates the output sequence y(n) of a linear-time invariant (LTI) system, with the input sequence x(n) and the unit sample sequence h(n), as shown in Fig. 1.convolution sum for discrete-time LTI systems and the convolution integral for continuous-time LTI systems. TRANSPARENCY 4.9 Evaluation of the convolution sum for an input that is a unit step and a system impulse response that is a decaying exponential for n > 0. The output of a discrete time LTI system is completely determined by the input and the system's response to a unit impulse. Figure 4.2.1 4.2. 1: We can determine the system's output, y[n] y [ n], if we know the system's impulse response, h[n] h [ n], and the input, x[n] x [ n]. The output for a unit impulse input is called the impulse response.Discrete Convolution • In the discrete case s(t) is represented by its sampled values at equal time intervals s j • The response function is also a discrete set r k – r 0 tells what multiple of the input signal in channel j is copied into the output channel j – r 1 tells what multiple of input signal j is copied into the output channel j+1Signal & System: Tabular Method of Discrete-Time Convolution Topics discussed:1. Tabulation method of discrete-time convolution.2. Example of the tabular met...Convolutions De nition/properties Convolution theorem Transfer function, Laplace vs. time space solutions 1 Introduction (what is the goal?) A car traveling on a road is, in its simplest form, a mass on a set of springs (the shocks). Bumps on the road apply a force that perturbs the car. A (very) simple model might take ….

Discrete time convolution is not simply a mathematical construct, it is a roadmap for how a discrete system works. This becomes especially useful when designing or implementing systems in discrete time such as digital filters and others which you may need to implement in embedded systems.The behavior of a linear, time-invariant discrete-time system with input signal x [n] and output signal y [n] is described by the convolution sum. The signal h [n], assumed known, is the response of the system to a unit-pulse input. The convolution summation has a simple graphical interpretation. 1, and for all time shifts k, then the system is called time-invariant or shift-invariant. A simple interpretation of time-invariance is that it does not matter when an input is applied: a delay in applying the input results in an equal delay in the output. 2.1.5 Stability of linear systems 1, and for all time shifts k, then the system is called time-invariant or shift-invariant. A simple interpretation of time-invariance is that it does not matter when an input is applied: a delay in applying the input results in an equal delay in the output. 2.1.5 Stability of linear systems A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (the Fourier transform of the sampling …(We will discuss in discrete time domain only.) where x[n] is input signal, h[n] is impulse response, and y[n] is output. * denotes convolution. Notice that we multiply the terms of x[k] by the terms of a time-shifted h[n] and add them up. The keystone of understanding convolution is lying behind impulse response and impulse decomposition.DSP - Operations on Signals Convolution. The convolution of two signals in the time domain is equivalent to the multiplication of their representation in frequency domain. Mathematically, we can write the convolution of two signals as. y(t) = x1(t) ∗ x2(t) = ∫∞ − ∞x1(p). x2(t − p)dp.More seriously, signals are functions of time (continuous-time signals) or sequences in time (discrete-time signals) that presumably represent quantities of interest. Systems are operators that accept a given signal (the input signal) and produce a new signal (the output signal). Of course, this is an abstraction of the processing of a signal.This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Classification of Signals”. 1. What is single-valued function? a) Single value for all instants of time. b) Unique value for every instant of time. c) A single pattern is followed by after ‘t’ intervals. d) Different pattern of values is followed by ... Discrete time convolution, roles in continuous time and discrete time. As with the continuous-time Four ier transform, the discrete-time Fourier transform is a complex-valued func-tion whether or not the sequence is real-valued. Furthermore, as we stressed in Lecture 10, the discrete-time Fourier transform is always a periodic func-tion of fl., Nov 23, 2022 · Convolution of 2 discrete time signals. My background: until very recently in my studies I was dealing with analog systems and signals and now we are being taught discrete signals. Suppose the impulse response of a discrete linear and time invariant system is h ( n) = u ( n) Find the output signal if the input signal is x ( n) = u ( n − 1 ... , Spring 2008 Discrete-Time Convolution Linear Systems and SignalsLecture 8. Linear Time-Invariant System • Any linear time-invariant system (LTI) system, continuous-time or discrete-time, can be uniquely characterized by its • Impulse response: response of system to an impulse • Frequency response: response of system to a complex exponential e j 2 p f for all possible frequencies f ..., A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra , and in the design and implementation of finite impulse response filters in signal processing. , The identity under convolution is the unit impulse. (t0) gives x 0. u (t) gives R t 1 x dt. Exercises Prove these. Of the three, the first is the most difficult, and the second the easiest. 4 Time Invariance, Causality, and BIBO Stability Revisited Now that we have the convolution operation, we can recast the test for time invariance in a new ... , 4.3: Discrete Time Convolution. Convolution is a concept that extends to all systems that are both linear and time-invariant (LTI). It will become apparent in this discussion that this condition is necessary by demonstrating how linearity and time-invariance give rise to convolution. 4.4: Properties of Discrete Time Convolution., The convolution theorem states that convolution in the time domain is equivalent to multiplication in the frequency domain. The frequency domain can also be used to improve the execution time of convolutions. Using the FFT algorithm, signals can be transformed to the frequency domain, multiplied, and transformed back to the time domain. For ..., May 22, 2022 · Introduction. This module relates circular convolution of periodic signals in one domain to multiplication in the other domain. You should be familiar with Discrete-Time Convolution (Section 4.3), which tells us that given two discrete-time signals \(x[n]\), the system's input, and \(h[n]\), the system's response, we define the output of the system as , w = conv (u,v) returns the convolution of vectors u and v. If u and v are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials. w = conv (u,v,shape) returns a subsection of the convolution, as specified by shape . For example, conv (u,v,'same') returns only the central part of the convolution, the ..., Source. Fullscreen. The output signal of an LTI (linear time-invariant) system with the impulse response is given by the convolution of the input signal with the impulse response of the system. Convolution is defined as . In this example, the input is a rectangular pulse of width and , which is the impulse response of an RC low‐pass filter., The identity under convolution is the unit impulse. (t0) gives x 0. u (t) gives R t 1 x dt. Exercises Prove these. Of the three, the first is the most difficult, and the second the easiest. 4 Time Invariance, Causality, and BIBO Stability Revisited Now that we have the convolution operation, we can recast the test for time invariance in a new ... , 10.1: Signal Sampling. This module introduces sampling of a continuous time signal to produce a discrete time signal, including a computation of the spectrum of the sampled signal and a discussion of its implications for reconstruction. 10.2: Sampling Theorem. This module builds on the intuition developed in the sampling module to discuss the ..., Discrete-Time Convolution Convolution is such an effective tool that can be utilized to determine a linear time-invariant (LTI) system’s output from an input and the impulse response knowledge. Given two discrete time signals x[n] and h[n], the convolution is defined by , we know that the definition of DTFT is. X(jω) = ∑n=−∞+∞ x[n]e−jωn X ( j ω) = ∑ n = − ∞ + ∞ x [ n] e − j ω n. Multiplication in Time domain will be convolution in DTFT. If we take the DTFT of anu[n] a n u [ n] we have. 1 1 − ae−jω 1 1 − a e − j ω. and DTFT of sin(ω0n)u[n] sin ( ω 0 n) u [ n] will be. π j ∑l ..., The unit sample sequence plays the same role for discrete-time signals and systems that the unit impulse function (Dirac delta function) does for continuous-time signals and systems. For convenience, we often refer to the unit sample sequence as a discrete-time impulse or simply as an impulse. It is important to note that a discrete-time impulse, The Discrete Fourier Transform (DFT) Midterm Exam 16 Linear Filtering with the DFT 17 Spectral ... FFT Algorithms 20 The Goertzel Algorithm and the Chirp Transform 21 Short-time Fourier Analysis 22 Modulated Filter Bank 23 Caruso’s Orchestra Final Exam Course Info Instructor Prof. Alan V. Oppenheim; Departments Electrical Engineering and ..., Discretion is a police officer’s option to use his judgment to interpret the law as it applies to misdemeanor crimes. The laws that apply to felony crimes, such as murder, are black and white., Answer: A. Clarification: The tools used in a graphical method of finding convolution of discrete time signals are basically plotting, shifting, folding, multiplication and addition. These are taken in the order in the graphs. Both the signals are plotted, one of them is shifted, folded and both are again multiplied and added., In a discrete-time system, the input-output relationship of a signal delay system is expressed as: y (l T) ... The simplified block diagram for a FDF is shown in Fig. 2, which output for a no causal FIR FDF filter …, Convolution of discrete-time signals Causal LTI systems with causal inputs Discrete convolution: an example The unit pulse response Let us consider a discrete-time LTI system y[n] = Snx[n]o and use the unit pulse δ[n] = 1, n = 0 0, n 6 = 0 as input. δ[n] 0 1 n Let us define the unit pulse response of S as the corresponding output: h[n] = Snδ[n]o, 9: Discrete Time Fourier Transform (DTFT), convolution representation of a discrete-time LTI system. This name comes from the fact that a summation of the above form is known as the convolution of two signals, in this case x[n] and h[n] = S n δ[n] o. Maxim Raginsky Lecture VI: Convolution representation of discrete-time systems, Visual comparison of convolution, cross-correlation, and autocorrelation.For the …, gives the convolution with respect to n of the expressions f and g. DiscreteConvolve [ f , g , { n 1 , n 2 , … } , { m 1 , m 2 , … gives the multidimensional convolution. , Discrete Time Convolution Neso Academy 2.25M subscribers Join Subscribe 2.2K Share 262K views 5 years ago Signals and Systems Signal & System: Discrete Time Convolution Topics discussed: 1...., 10 years ago. Convolution reverb does indeed use mathematical convolution as seen here! First, an impulse, which is just one tiny blip, is played through a speaker into a space (like a cathedral or concert hall) so it echoes. (In fact, an impulse is pretty much just the Dirac delta equation through a speaker!), Viewed 38 times. 1. h[n] = (8 9)n u[n − 3] h [ n] = ( 8 9) n u [ n − 3] And the function is: x[n] ={2 0 if 0 ≤ n ≤ 9, else. x [ n] = { 2 if 0 ≤ n ≤ 9, 0 else. In order to find the convolution sum y[n] = x[n] ∗ h[n] y [ n] = x [ n] ∗ h [ n]: y[n] = ∑n=−∞+∞ x[n] ⋅ h[k − n] y [ n] = ∑ n = − ∞ + ∞ x [ n] ⋅ h ..., The output of a discrete time LTI system is completely determined by the input and the system's response to a unit impulse. Figure 4.2.1 4.2. 1: We can determine the system's output, y[n] y [ n], if we know the system's impulse response, h[n] h [ n], and the input, x[n] x [ n]. The output for a unit impulse input is called the impulse response., 1 Answer. Sorted by: 1. The multiplication of the two unit step sequences u[k] ⋅ u[−n + k − 1] u [ k] ⋅ u [ − n + k − 1] is only non-zero if both sequences are non-zero. This means that the condition k ≥ 0 k ≥ 0 as well as the condition k ≥ n + 1 k ≥ n + 1 must be satisfied. So you have two cases: for n <= −1 n <= − 1 ..., Discrete-Time Convolution - Wolfram Demonstrations Project The convolution of two discretetime signals and is defined as The left column shows and below over The right column shows the product over and below the result over, 10.1: Signal Sampling. This module introduces sampling of a continuous time signal to produce a discrete time signal, including a computation of the spectrum of the sampled signal and a discussion of its implications for reconstruction. 10.2: Sampling Theorem. This module builds on the intuition developed in the sampling module to discuss the ..., The discrete-time Fourier transform X (ω) of a discrete-time sequence x(n) x ( n) represents the frequency content of the sequence x(n) x ( n). Therefore, by taking the Fourier transform of the discrete-time sequence, the sequence is decomposed into its frequency components. For this reason, the DTFT X (ω) is also called the signal spectrum., Discrete Convolution • In the discrete case s(t) is represented by its sampled values at equal time intervals s j • The response function is also a discrete set r k – r 0 tells what multiple of the input signal in channel j is copied into the output channel j – r 1 tells what multiple of input signal j is copied into the output channel j+1