DOCSLIB.ORG

Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 1 News Homework #5 . Due this week . Submit via canvas Coding Problem #4 . Due this week . Submit via canvas Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 2 Exam 1 Grades The class did exceedingly well . Mean: 89.3 . Median: 91.5 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 3 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 4 Sampling Discrete-Time Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 5 Sampling Sampling Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 6 Discrete-Time Fourier Transform Example: Consider the DTFT signal = ∞ + + 2 4 − 8 �+ − 2 − − 2 =−∞ 2 4 . Sketch the magnitude of − .− 2 − − − 2 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 7 Discrete-Time Fourier Transform Example: Consider the DTFT signal = ∞ + + 2 4 − 8 �+ − 2 − − 2 =−∞ 2 4 . Sketch the phase of . − − 2 − − − 2 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 8 Sampling 2 Ω Ω ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 9 Sampling 2 Ω Ω 1 2 Ω ∗ Ω Ω 2 ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 10 Sampling 2 Ω Ω 1 ∞ � Ω − Ω =−∞ ℱ 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 2 1 1 2 3 Ω −3 −2 − 2 3 − Ωs − Ωs − Ωs Ωs Ωs Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 11 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 12 Sampling Question: Can I preserve all information when I sample? . Yes! Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 13 Sampling Question: How fast do I sample to preserve information? Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 14 Sampling Question: How fast do I sample to preserve information? = 1 Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 15 Sampling Question: How fast do I sample to preserve information? = 1 Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 16 Sampling Question: How fast do I sample to preserve information? = 1 Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 17 Sampling Question: How fast do I sample to preserve information? = 1 Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 18 Sampling Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2 Ωs Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 19 Sampling Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 20 Sampling Question: How fast do I sample to preserve information? . We need to sample twice as fast as the maximum frequency > 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max = 2 <- Nyquist Rate ΩN Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 21 Example Example: Consider the signal = cos . What is the Nyquist rate? 5 . Sketch the Fourier transform of the sampled signal when = 12 Ω Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 22 Example Example: Consider the frequency signal = + . What is the Nyquist rate? Ω 5 − − 5 . Sketch the Fourier transform of the sampled signal when = 20 Ω Ωs Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 23 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 24 Sampling Question: How do I return to continuous–time? > 2 Nyquist-Shannon f > 2f Sampling Theorem Ωs Ωmax s max = 2 <- Nyquist Rate ΩN Ωmax Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 25 Sampling Question: How do I return to continuous–time? . Filter: Low pass filter to keep /2 /2 . Amplify: Amplify signal by Ω ≤ Ω ≤ Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 26 Sampling Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 27 Sampling Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω = = 1.5708 2 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 28 Sampling Question: How do I return to continuous–time? . Apply a low-pass reconstruction filter ◊ Cut-off frequency: /2 ◊ Gain: Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 29 Sampling Question: What is happening when I multiply in frequency? + /2 /2 Ω Ω Ω − Ω − Ω = = 1.5708 2 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 30 Sampling Question: What is happening when I multiply in frequency? rect Ω Ω Ω Ω = = 1.5708 2 Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 31 Sampling Question: What is happening when I multiply in frequency? rect 2 /2 Ω Ω Ω /2 sinc /2 Ω ∗ Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 32 Sampling Question: What is happening when I multiply in frequency? rect 2 /2 Ω Ω Ω /2 sinc /2 = sinc /2 Ω ∗ Ω ∗ Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 33 Sampling Question: What is happening when I multiply in frequency? rect 2 /2 Ω Ω Ω sinc /2 ∗ Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 34 Sampling Question: What is happening when I multiply in frequency? rect 2 /2 Ω Ω Ω sinc /2 ∗ Ω Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 35 Reconstruction Consider a cosine Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 36 Reconstruction Consider a cosine – Sampled at = Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 37 Reconstruction Consider a cosine – Low pass filter to keep / to / − Low Pass Reconstruction Filter Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier
Load more

Recommended publications
  • Mathematical Basics of Bandlimited Sampling and Aliasing
    Signal Processing Mathematical basics of bandlimited sampling and aliasing Modern applications often require that we sample analog signals, convert them to digital form, perform operations on them, and reconstruct them as analog signals. The important question is how to sample and reconstruct an analog signal while preserving the full information of the original. By Vladimir Vitchev o begin, we are concerned exclusively The limits of integration for Equation 5 T about bandlimited signals. The reasons are specified only for one period. That isn’t a are both mathematical and physical, as we problem when dealing with the delta func- discuss later. A signal is said to be band- Furthermore, note that the asterisk in Equa- tion, but to give rigor to the above expres- limited if the amplitude of its spectrum goes tion 3 denotes convolution, not multiplica- sions, note that substitutions can be made: to zero for all frequencies beyond some thresh- tion. Since we know the spectrum of the The integral can be replaced with a Fourier old called the cutoff frequency. For one such original signal G(f), we need find only the integral from minus infinity to infinity, and signal (g(t) in Figure 1), the spectrum is zero Fourier transform of the train of impulses. To the periodic train of delta functions can be for frequencies above a. In that case, the do so we recognize that the train of impulses replaced with a single delta function, which is value a is also the bandwidth (BW) for this is a periodic function and can, therefore, be the basis for the periodic signal.
    [Show full text]
  • Efficient Supersampling Antialiasing for High-Performance Architectures
    Efficient Supersampling Antialiasing for High-Performance Architectures TR91-023 April, 1991 Steven Molnar The University of North Carolina at Chapel Hill Department of Computer Science CB#3175, Sitterson Hall Chapel Hill, NC 27599-3175 This work was supported by DARPA/ISTO Order No. 6090, NSF Grant No. DCI- 8601152 and IBM. UNC is an Equa.l Opportunity/Affirmative Action Institution. EFFICIENT SUPERSAMPLING ANTIALIASING FOR HIGH­ PERFORMANCE ARCHITECTURES Steven Molnar Department of Computer Science University of North Carolina Chapel Hill, NC 27599-3175 Abstract Techniques are presented for increasing the efficiency of supersampling antialiasing in high-performance graphics architectures. The traditional approach is to sample each pixel with multiple, regularly spaced or jittered samples, and to blend the sample values into a final value using a weighted average [FUCH85][DEER88][MAMM89][HAEB90]. This paper describes a new type of antialiasing kernel that is optimized for the constraints of hardware systems and produces higher quality images with fewer sample points than traditional methods. The central idea is to compute a Poisson-disk distribution of sample points for a small region of the screen (typically pixel-sized, or the size of a few pixels). Sample points are then assigned to pixels so that the density of samples points (rather than weights) for each pixel approximates a Gaussian (or other) reconstruction filter as closely as possible. The result is a supersampling kernel that implements importance sampling with Poisson-disk-distributed samples. The method incurs no additional run-time expense over standard weighted-average supersampling methods, supports successive-refinement, and can be implemented on any high-performance system that point samples accurately and has sufficient frame-buffer storage for two color buffers.
    [Show full text]
  • Detection and Estimation Theory Introduction to ECE 531 Mojtaba Soltanalian- UIC the Course
    Detection and Estimation Theory Introduction to ECE 531 Mojtaba Soltanalian- UIC The course Lectures are given Tuesdays and Thursdays, 2:00-3:15pm Office hours: Thursdays 3:45-5:00pm, SEO 1031 Instructor: Prof. Mojtaba Soltanalian office: SEO 1031 email: [email protected] web: http://msol.people.uic.edu/ The course Course webpage: http://msol.people.uic.edu/ECE531 Textbook(s): * Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory, by Steven M. Kay, Prentice Hall, 1993, and (possibly) * Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory, by Steven M. Kay, Prentice Hall 1998, available in hard copy form at the UIC Bookstore. The course Style: /Graduate Course with Active Participation/ Introduction Let’s start with a radar example! Introduction> Radar Example QUIZ Introduction> Radar Example You can actually explain it in ten seconds! Introduction> Radar Example Applications in Transportation, Defense, Medical Imaging, Life Sciences, Weather Prediction, Tracking & Localization Introduction> Radar Example The strongest signals leaking off our planet are radar transmissions, not television or radio. The most powerful radars, such as the one mounted on the Arecibo telescope (used to study the ionosphere and map asteroids) could be detected with a similarly sized antenna at a distance of nearly 1,000 light-years. - Seth Shostak, SETI Introduction> Estimation Traditionally discussed in STATISTICS. Estimation in Signal Processing: Digital Computers ADC/DAC (Sampling) Signal/Information Processing Introduction> Estimation The primary focus is on obtaining optimal estimation algorithms that may be implemented on a digital computer. We will work on digital signals/datasets which are typically samples of a continuous-time waveform.
    [Show full text]
  • Z-Transformation - I
    Digital signal processing: Lecture 5 z-transformation - I Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/1 Review of last lecture • Fourier transform & inverse Fourier transform: – Time domain & Frequency domain representations • Understand the “true face” (latent factor) of some physical phenomenon. – Key points: • Definition of Fourier transformation. • Definition of inverse Fourier transformation. • Condition for a signal to have FT. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/2 Review of last lecture • The sampling theorem – If the highest frequency component contained in an analog signal is f0, the sampling frequency (the Nyquist rate) should be 2xf0 at least. • Frequency response of an LTI system: – The Fourier transformation of the impulse response – Physical meaning: frequency components to keep (~1) and those to remove (~0). – Theoretic foundation: Convolution theorem. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/3 Topics of this lecture Chapter 6 of the textbook • The z-transformation. • z－変換 • Convergence region of z- • z－変換の収束領域 transform. －変換とフーリエ変換 • Relation between z- • z transform and Fourier • z－変換と差分方程式 transform. • 伝達関数 or システム • Relation between z- 関数 transform and difference equation. • Transfer function (system function). Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/4 z-transformation（z-変換） • Laplace transformation is important for analyzing analog signal/systems. • z-transformation is important for analyzing discrete signal/systems. • For a given signal x(n), its z-transformation is defined by ∞ Z[x(n)] = X (z) = ∑ x(n)z−n (6.3) n=0 where z is a complex variable. Produced by Qiangfu Zhao (Since 1995), All rights reserved © DSP-Lec05/5 One-sided z-transform and two-sided z-transform • The z-transform defined above is called one- sided z-transform (片側ｚ-変換）.
    [Show full text]
  • Basics on Digital Signal Processing
    Basics on Digital Signal Processing z - transform - Digital Filters Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras Outline of the Lecture 1. The z-transform 2. Properties 3. Examples 4. Digital filters 5. Design of IIR and FIR filters 6. Linear phase 2/31 z-Transform 2 N 1 N 1 j nk n X (k) x(n) e N X (z) x(n) z n0 n0 Time to frequency Time to z-domain (complex) Transformation tool is the Transformation tool is complex wave z e j e j j j2k / N e e With amplitude ρ changing(?) with time With amplitude |ejω|=1 The z-transform is more general than the DFT 3/31 z-Transform For z=ejω i.e. ρ=1 we work on the N 1 unit circle X (z) x(n) z n And the z-transform degenerates n0 into the Fourier transform. I -z z=R+jI |z|=1 R The DFT is an expression of the z-transform on the unit circle. The quantity X(z) must exist with finite value on the unit circle i.e. must posses spectrum with which we can describe a signal or a system. 4/31 z-Transform convergence We are interested in those values of z for which X(z) converges. This region should contain the unit circle. I Why is it so? |a| N 1 N 1 X (z) x(n) zn x(n) (1/zn ) n0 n0 R At z=0, X(z) diverges ROC The values of z for which X(z) diverges are called poles of X(z).
    [Show full text]
  • 3F3 – Digital Signal Processing (DSP)
    3F3 – Digital Signal Processing (DSP) Simon Godsill www-sigproc.eng.cam.ac.uk/~sjg/teaching/3F3 Course Overview • 11 Lectures • Topics: – Digital Signal Processing – DFT, FFT – Digital Filters – Filter Design – Filter Implementation – Random signals – Optimal Filtering – Signal Modelling • Books: – J.G. Proakis and D.G. Manolakis, Digital Signal Processing 3rd edition, Prentice-Hall. – Statistical digital signal processing and modeling -Monson H. Hayes –Wiley • Some material adapted from courses by Dr. Malcolm Macleod, Prof. Peter Rayner and Dr. Arnaud Doucet Digital Signal Processing - Introduction • Digital signal processing (DSP) is the generic term for techniques such as filtering or spectrum analysis applied to digitally sampled signals. • Recall from 1B Signal and Data Analysis that the procedure is as shown below: • is the sampling period • is the sampling frequency • Recall also that low-pass anti-aliasing filters must be applied before A/D and D/A conversion in order to remove distortion from frequency components higher than Hz (see later for revision of this). • Digital signals are signals which are sampled in time (“discrete time”) and quantised. • Mathematical analysis of inherently digital signals (e.g. sunspot data, tide data) was developed by Gauss (1800), Schuster (1896) and many others since. • Electronic digital signal processing (DSP) was first extensively applied in geophysics (for oil-exploration) then military applications, and is now fundamental to communications, mobile devices, broadcasting, and most applications of signal and image processing. There are many advantages in carrying out digital rather than analogue processing; among these are flexibility and repeatability. The flexibility stems from the fact that system parameters are simply numbers stored in the processor.
    [Show full text]
  • Super-Sampling Anti-Aliasing Analyzed
    Super-sampling Anti-aliasing Analyzed Kristof Beets Dave Barron Beyond3D [email protected] Abstract - This paper examines two varieties of super-sample anti-aliasing: Rotated Grid Super- Sampling (RGSS) and Ordered Grid Super-Sampling (OGSS). RGSS employs a sub-sampling grid that is rotated around the standard horizontal and vertical offset axes used in OGSS by (typically) 20 to 30°. RGSS is seen to have one basic advantage over OGSS: More effective anti-aliasing near the horizontal and vertical axes, where the human eye can most easily detect screen aliasing (jaggies). This advantage also permits the use of fewer sub-samples to achieve approximately the same visual effect as OGSS. In addition, this paper examines the fill-rate, memory, and bandwidth usage of both anti-aliasing techniques. Super-sampling anti-aliasing is found to be a costly process that inevitably reduces graphics processing performance, typically by a substantial margin. However, anti-aliasing’s posi- tive impact on image quality is significant and is seen to be very important to an improved gaming experience and worth the performance cost. What is Aliasing? digital medium like a CD. This translates to graphics in that a sample represents a specific moment as well as a Computers have always strived to achieve a higher-level specific area. A pixel represents each area and a frame of quality in graphics, with the goal in mind of eventu- represents each moment. ally being able to create an accurate representation of reality. Of course, to achieve reality itself is impossible, At our current level of consumer technology, it simply as reality is infinitely detailed.
    [Show full text]
  • Discrete - Time Signals and Systems
    Discrete - Time Signals and Systems Sampling – II Sampling theorem & Reconstruction Yogananda Isukapalli Sampling at diffe- -rent rates From these figures, it can be concluded that it is very important to sample the signal adequately to avoid problems in reconstruction, which leads us to Shannon’s sampling theorem 2 Fig:7.1 Claude Shannon: The man who started the digital revolution Shannon arrived at the revolutionary idea of digital representation by sampling the information source at an appropriate rate, and converting the samples to a bit stream Before Shannon, it was commonly believed that the only way of achieving arbitrarily small probability of error in a communication channel was to 1916-2001 reduce the transmission rate to zero. All this changed in 1948 with the publication of “A Mathematical Theory of Communication”—Shannon’s landmark work Shannon’s Sampling theorem A continuous signal xt( ) with frequencies no higher than fmax can be reconstructed exactly from its samples xn[ ]= xn [Ts ], if the samples are taken at a rate ffs ³ 2,max where fTss= 1 This simple theorem is one of the theoretical Pillars of digital communications, control and signal processing Shannon’s Sampling theorem, • States that reconstruction from the samples is possible, but it doesn’t specify any algorithm for reconstruction • It gives a minimum sampling rate that is dependent only on the frequency content of the continuous signal x(t) • The minimum sampling rate of 2fmax is called the “Nyquist rate” Example1: Sampling theorem-Nyquist rate x( t )= 2cos(20p t ), find the Nyquist frequency ? xt( )= 2cos(2p (10) t ) The only frequency in the continuous- time signal is 10 Hz \ fHzmax =10 Nyquist sampling rate Sampling rate, ffsnyq ==2max 20 Hz Continuous-time sinusoid of frequency 10Hz Fig:7.2 Sampled at Nyquist rate, so, the theorem states that 2 samples are enough per period.
    [Show full text]
  • CONVOLUTION: Digital Signal Processing .R .Hamming, W
    CONVOLUTION: Digital Signal Processing AN-237 National Semiconductor CONVOLUTION: Digital Application Note 237 Signal Processing January 1980 Introduction As digital signal processing continues to emerge as a major Decreasing the pulse width while increasing the pulse discipline in the field of electrical engineering, an even height to allow the area under the pulse to remain constant, greater demand has evolved to understand the basic theo- Figure 1c, shows from eq(1) and eq(2) the bandwidth or retical concepts involved in the development of varied and spectral-frequency content of the pulse to have increased, diverse signal processing systems. The most fundamental Figure 1d. concepts employed are (not necessarily listed in the order Further altering the pulse to that of Figure 1e provides for an [ ] of importance) the sampling theorem 1 , Fourier transforms even broader bandwidth, Figure 1f. If the pulse is finally al- [ ][] 2 3, convolution, covariance, etc. tered to the limit, i.e., the pulsewidth being infinitely narrow The intent of this article will be to address the concept of and its amplitude adjusted to still maintain an area of unity convolution and to present it in an introductory manner under the pulse, it is found in 1g and 1h the unit impulse hopefully easily understood by those entering the field of produces a constant, or ``flat'' spectrum equal to 1 at all digital signal processing. frequencies. Note that if ATe1 (unit area), we get, by defini- It may be appropriate to note that this article is Part II (Part I tion, the unit impulse function in time.
    [Show full text]
  • The Nyquist Sampling Rate for Spiraling Curves 11
    THE NYQUIST SAMPLING RATE FOR SPIRALING CURVES PHILIPPE JAMING, FELIPE NEGREIRA & JOSE´ LUIS ROMERO Abstract. We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of under- sampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis. 1. Introduction 1.1. The mobile sampling problem. In this article, we consider the reconstruction of a compactly supported function from samples of its Fourier transform taken along certain curves, that we call spiraling. This problem is relevant, for example, in magnetic resonance imaging (MRI), where the anatomy and physiology of a person are captured by moving sensors. The Fourier sampling problem is equivalent to the sampling problem for bandlimited functions - that is, functions whose Fourier transform are supported on a given compact set. The most classical setting concerns functions of one real variable with Fourier transform supported on the unit interval [ 1/2, 1/2], and sampled on a grid ηZ, with η > 0. The sampling rate η determines whether− every bandlimited function can be reconstructed from its samples: reconstruction fails if η > 1 and succeeds if η 6 1 [42]. The transition value η = 1 is known as the Nyquist sampling rate, and it is the benchmark for all sampling schemes: modern sampling strategies that exploit the particular structure of a certain class of signals are praised because they achieve sub-Nyquist sampling rates.
    [Show full text]
  • 2.161 Signal Processing: Continuous and Discrete Fall 2008
    MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete 1 Sampling and the Discrete Fourier Transform 1 Sampling Consider a continuous function f(t) that is limited in extent, T1 · t < T2. In order to process this function in the computer it must be sampled and represented by a ¯nite set of numbers. The most common sampling scheme is to use a ¯xed sampling interval ¢T and to form a sequence of length N: ffng (n = 0 : : : N ¡ 1), where fn = f(T1 + n¢T ): In subsequent processing the function f(t) is represented by the ¯nite sequence ffng and the sampling interval ¢T . In practice, sampling occurs in the time domain by the use of an analog-digital (A/D) converter. The mathematical operation of sampling (not to be confused with the physics of sampling) is most commonly described as a multiplicative operation, in which f(t) is multiplied by a Dirac comb sampling function s(t; ¢T ), consisting of a set of delayed Dirac delta functions: X1 s(t; ¢T ) = ±(t ¡ n¢T ): (1) n=¡1 ? We denote the sampled waveform f (t) as X1 ? f (t) = s(t; ¢T )f(t) = f(t)±(t ¡ n¢T ) (2) n=¡1 ? as shown in Fig. 1. Note that f (t) is a set of delayed weighted delta functions, and that the waveform must be interpreted in the distribution sense by the strength (or area) of each component impulse.
    [Show full text]
  • Efficient Multidimensional Sampling
    EUROGRAPHICS 2002 / G. Drettakis and H.-P. Seidel Volume 21 (2002 ), Number 3 (Guest Editors) Efficient Multidimensional Sampling Thomas Kollig and Alexander Keller Department of Computer Science, Kaiserslautern University, Germany Abstract Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized con- cept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jittered sampling and N-rooks sampling at the same time. The faster convergence and improved anti-aliasing are demonstrated by numerical experiments. Categories and Subject Descriptors (according to ACM CCS): G.3 [Probability and Statistics]: Prob- abilistic Algorithms (including Monte Carlo); I.3.2 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism. 1. Introduction general concept of stratification than just joining jit- tered and Latin hypercube sampling. Since our sam- Many rendering tasks are given in integral form and ples are highly correlated and satisfy a minimum dis- usually the integrands are discontinuous and of high tance property, noise artifacts are attenuated much 22 dimension, too. Since the Monte Carlo method is in- more efficiently and anti-aliasing is improved. dependent of dimension and applicable to all square- integrable functions, it has proven to be a practical tool for numerical integration. It relies on the point 2. Monte Carlo Integration sampling paradigm and such on sample placement. In- The Monte Carlo method of integration estimates the creasing the uniformity of the samples is crucial for integral of a square-integrable function f over the s- the efficiency of the stochastic method and the level dimensional unit cube by of noise contained in the rendered images.
    [Show full text]