測試信號英文版課件：Chapter3 Discrete-Time Signals in the Transform Domain_Lec2_part1

1、Discrete Fourier TransformDefinitionDFT Computation Using MATLABRelation between DTFT and DFT and their inversesA simple DFT application-linear convolution12Linear Convolution Using DTFT1) Compute the DTFTs and of the sequences xn and hn, respectively2) Form the DTFT3) Compute the IDFT yn of xnhnynD

2、TFTDTFTIDTFT3Problems Needed to be Solved in Practical Applications4Discrete Fourier TransformDefinition - The simplest relation between a length-N sequence xn, defined for , and its DTFT , it is obtained by uniformly sampling on the w-axis between at ,From the definition of the DTFT we thus have5有限

3、長序列的DFT就是序列的傅里葉變換以 為間隔的抽樣值 Discrete Fourier TransformDiscretization in Time-domain and Frequency-domain7Time-domainSamplingFrequency-domainSamplingDTFTDFTDiscrete Fourier TransformNote: Xk is also a length-N sequence in the frequency domainThe sequence Xk is called the discrete Fourier transform (DF

4、T) of the sequence xnUsing the notation the DFT is usually expressed as:8Discrete Fourier TransformThe inverse discrete Fourier transform (IDFT) is given by9Discrete Fourier TransformDFT: analysis equationIDFT: synthesis equationtime domainfrequency domain10Transformation method NameTime domaintrans

5、form domainTransformation method Continuous-time Fourier Transform (CTFT)Discrete-Time Fourier Transform (DTFT)Discrete Fourier Transform (DFT)11Discrete Fourier TransformExample - Consider the length-N sequenceIts N-point DFT is given by 12Discrete Fourier TransformExample - Consider the length-N s

6、equenceIts N-point DFT is given by 13Discrete Fourier TransformExample - Consider the length-N sequence defined forUsing a trigonometric identity we can write14Discrete Fourier TransformThe N-point DFT of gn is thus given by15Discrete Fourier TransformMaking use of the identitywe getr an integer16DF

7、T Computation Using MATLABExample figure below shows the DFT and the DTFT of the sequenceindicates DFT samples17Xk=8, k=38, k=130, otherwiseDFT Computation Using MATLAB18DFT Computation Using MATLAB19DFT Computation Using MATLABThe functions to compute the DFT and the IDFT are fft and ifftThese func

8、tions make use of Fast Fourier Transform (FFT) algorithms which are computationally highly efficient compared to the direct computationR=(computation complexity of FFT)/ (computation complexity of DFT)=(Nlog2N)/(N(N-1) log2(N)/NE.g., N=1024, R=1/200, 2021有限長序列的DFT就是序列的傅里葉變換以 為間隔的抽樣值 Discrete Fourier

9、 TransformDTFT from DFT by Interpolation The N-point DFT Xk of a length-N sequence xn is simply the frequency samples of its DTFT evaluated at N uniformly spaced frequency pointsGiven the N-point DFT Xk of a length-N sequence xn, its DTFT can be uniquely determined from Xk 23DTFT from DFT by Interpo

10、lationThus24DTFT from DFT by InterpolationIt can readily be shown that25Sampling the DTFTConsider a length-M sequence xn with a DTFTWe sample at N equally spaced points , developing the N frequency samplesThese N frequency samples can be considered as an N-point DFT Yk whose N-point IDFT is a length

11、-N sequence yn26Sampling the DTFTNowThusAn IDFT of Yk yields27Sampling the DTFTi.e.Making use of the identity28Sampling the DTFTwe arrive at the desired relationThus yn is obtained from xn by adding an infinite number of shifted replicas of xn, with each replica shifted by an integer multiple of N s

12、ampling instants, and observing the sum only for the interval29yn=xn+xn+8+xn-8,0=n=730Sampling the DTFTTo applyto finite-length sequences, we assume that the samples outside the specified range are zerosThus if xn is a length-M sequence with , then yn = xn for32Sampling the DTFTIf N M, there is a ti

13、me-domain aliasing of samples of xn in generating yn, and xn cannot be recovered from ynExample - Let By sampling its DTFT at , and then applying a 8-point IDFT to these samples, according to last slide, we arrive at the sequence yn given by 33Sampling the DTFTyn=xn+xn+8+xn-8,0=n N:41有限長序列的DFT就是序列的傅

14、里葉變換以 為間隔的抽樣值 Discrete Fourier TransformNumerical Computation of the DTFT Using the DFTDefine a new sequenceThen43Numerical Computation of the DTFT Using the DFTThus is essentially an M-point DFT of the length-M sequenceThe DFT can be computed very efficiently using the FFT algorithm if M is an inte

15、ger power of 2The function freqz employs this approach to evaluate the frequency response at a prescribed set of frequencies of a DTFT expressed as a rational function of 44工程上所遇到的信號，包括傳感器的輸出信號，大多是連續(xù)非周期信號，這種信號無論是在時域或頻域都是連續(xù)的，其波形和頻譜如下圖所示。0ax (t)t連續(xù)非周期信號時域波形和頻譜Digital spectrum analysis of continuous-tim