版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
1、Chapter 11Discrete Fourier Transform (DFT),Links the DFT and DTFT, DFS,Identifies the limited resolution of the DFT and problems such as smearing that arise from it,Applies the DFT to non-periodic and periodic signals,The fast DFT algorithm -FFT,Defines and interprets the DFT,The definition of DFT(D
2、iscrete Fourier Transform):,11.1 DFT BASICS,According to the definition, DFT used to calculate the spectrum of discrete periodic signals, in fact, any finite length signal can be seen as a cycle of the periodic signal; Infinite length signals must be truncated to finite length signals to calculate.,
3、1. Links the DFT and DTFT:,The Discrete Fourier transform (DFT) of x(n) is its N-point sampling DTFT transform, namely = 2k/N.,11.2 Links the DFT and other transform,Links the DTFT and DFT, between signal and IDFT,Non-periodic signal,DTFT spectrum,DFT spectrum within the signal window is the samplin
4、g form of the DTFT,The signals generated by the inverse DFT is part of the window.,2. Links the DTFT and Z transform:,Links the DFT and other transform,The z transform of finite sequence x (n) with length is N,The DTFT of finite sequence x (n) with length is N,The Discrete Time Fourier transform (DT
5、FT) of sequence x(n) is its z-transform values on the unit circle(z = ej ).,3. DFT and Z-transform:,Sampling the unit circle of as,Calculate X(z):,Links the DFT and other transform,The DFT is sampling values of its z transformations on the unit circle.,Links the DFT and other transform,The links DFT
6、, DTFT and the Fourier transform is described as,Time domain sampling Time-domain windowing Frequency-domain sampling,Links the DFT and other transform,Time-domain and frequency -domain are continuous, non-periodic,Time-domain is discrete, non-periodic; frequency -domain is continuous cycle,Time-dom
7、ain is discrete, non-periodic; frequency-domain is continuous, periodic (copy), to quantified and spectrum aliasing of errors,Time-domain vector multiplication, frequency -domain convolution,Sample in the time domain,Time-domain is discrete, non-periodic; frequency domain is continuous and periodic,
8、The time-domain is discrete, non-periodic; frequency-domain is continuous, periodic and frequency leakage (plus windows directly generated error),Time-domain vector multiplication, frequency domain convolution,Window in the time-domain,Frequency domain is discrete, non-, time domain is continuous, p
9、eriodic,Frequency-domain is discrete and periodic , time-domain is discrete, periodic,Multiply the frequency domain loss, time-domain convolution,Sample in the Frequency-domain,Aliasing in the spectrum and quantified window process caused the error, but the DFT sample values are very close to the or
10、iginal signal spectrum DFT is close to the original spectrum on the N sampling points exactly.,Links the DFT and other transform,5. The links DFT and DFS,Links the DFT and other transform,The DFT covers the frequency range of 0 to fs. Frequency sampling points are at intervals of fs / N. Frequency i
11、nterval is smaller, the resolution is better ; the interval is greater, the DFT resolution is worse.,DFT frequency interval (frequency resolution),DFT component X (k) located at:,k = N / 2, f is the Nyquist boundary of fs/2,. Thus, k = 0 N / 2 , DFT points carry all the necessary amplitude and phase
12、 information, the remaining points are the mirror copy of the important signal frequency, symmetry about k = N / 2. This is the magnitude of the DFT spectrums feature .,DFT example,Ex11.4(p444): The following figure shows the 40 seconds cosine wave, these two signals are added together and combined
13、with random noise to produced a signal x(t). Analysis its spectrum.,The signal contains two major frequency components, 1/16Hz and 3/8Hz. Now sampling fs = 6.4Hz, the digital frequency of signal:,x(n):,DFT example,The digital signals 256 sampling points (envelope) within 40 seconds with 6.4Hz sampli
14、ng rate:,n=0:255; x=cos(2*pi*n/102.4)+cos(6*pi*n/51.2)+0.8*(rand(1,256)-0.5) plot(n,x); axis(0 255 2.5 2.5);,The amplitude spectrum calculated by DFT,f=fft(x); plot(n,abs(f); axis(0 255 0 140);,K = N / 2 = 128 point symmetry,K = N / 2 = 128 point symmetry,K = N / 2 = 128 point symmetry,Amplitude spe
15、ctrum of 0 to 127,plot(n(1:128),abs(f(1:128); axis(0 128 0 140);,Amplitude spectrum of 0 to 20,plot(n(1:20),abs(f(1:20),Lower frequency cosine 1/16Hz, i.e. between k = 2 and k = 3.,Higher frequency cosine 3/8Hz, i.e. k = 15,The digital signals 512 sampling points within 80 seconds with 6.4Hz samplin
16、g rate :,n=0:511; x=cos(2*pi*n/102.4)+cos(6*pi*n/51.2)+0.8*(rand(1,256)-0.5) plot(n,x); axis(0 512 2.5 2.5);,The calculation of the DFT spectrum amplitude,f=fft(x); plot(n,abs(f); axis(0 512 0 300);,the amplitude spectrum of 040,plot(n(1:40),abs(f(1:40),Low frequency cosine 1/16 Hz, k = 5,High frequ
17、ency cosine 3/8 Hz , k = 15,DFT example,The DFT magnitude spectral envelope of Non-periodic signal will show the size of the change, but no clear peak. The DFT spectrum of periodic signal will appear narrow peak. These peaks lie in the harmonic frequencies.,The relationship of DFT and the periodic c
18、omponent of the signal,DFT example,example 11.8(p455): The figure shows a portion of Touch-Tone signal for the number “4” . The 1024 samples are collected at 8kHz.,The peaks in the magnitude of the DFT spectrum reveals that the signal has two components,700Hz and 1209Hz, respectively. So the key num
19、ber 4 is identified.,DFT example,example 11.10(p458): Digital white noise signal and its DFT magnitude spectrum as,Because there is no obvious peak, the signal is not periodic. In addition, because the contribution of all frequencies of the white noise signal are equal, all the amplitude spectrum ar
20、e approximately flat.,The finite set of time samples selected by a DFT is often said to lie within the DFT window,11.4 DFT Window effects,Spectral leakage,there is a clear difference between the spectrum of the truncated sequence and the spectrum of the original sequence.,Spectral interference,32 sa
21、mples,128 samples,Spectrum of sinusoid using rectangular windows of different lengths.,The window length is shorter, the peak is wider.,64 samples ( windows length),Select sinusoidal peak side lobe is less than the main lobe at least 40dB in order to approximate. Hamming window, Blackman window and
22、Kaiser window can achieve this goal. Use a longer window to improve the accuracy of signal analysis in the practical application .,Spectrum of sinusoid using nonrectangular windows,The spectrogram plots frequency against time so that each vertical slice of diagram contains the DFT magnitudes for one
23、 window of time.,11.5 Spectrograms,Spectrogram of phrase,Spectrograms,Humpback whale sounds Spectrogram,Spectrograms,Wider spacing indicates higher base frequency,Spectrogram of bird song,Spectrograms,Frequency changes rapidly,Spectrogram of Echolocation sounds,A wide frequency range,Spectrograms,Sp
24、ectrogram of Didgeridoo sound,A strong tone,Spectrograms,Big Ben bell audio spectrum,Low-frequency tone,Spectrograms,Telephone busy signal spectrum,Dual Tone Multi Frequency(DTMF) Repeat sequences of the two mono,Spectrograms,Motorcycle passing spectrogram,Doppler shift,Spectrograms,11.7 FFT basics,
25、Visible that, each calculation of X (k) need N complex multiplications and N-1 complex additions. Calculate all of the X (k) requires N2 complex multiplications and N(N-1)N2 complex addition, and with the increasing of N, it shows a nonlinear increasing.,The basic way to reduce the computational com
26、plexity,1、 Decomposition A combination of the larger N points DFT is decomposed into a number of small points of DFT can reduce the computational complexity. This is also the basic technical means of the FFT.,The basic way to reduce the computational complexity,N/4,N/2,N,N2/2,(N/2)2,(N/2)2,N2,N/2,N/4,N/4,N/4,2 Use the characteristics of the rotation factor WN,symmetrical characteristic:,The basic way to reduce the computational complexity,Periodicity,Special value,The basic
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- Unit6单元知识点归纳总结-人教版七年级英语下册
- 无理数第1课时学案七年级数学上册
- 水果批发入股合同范本
- 地下防水维修合同
- 财务顾问协议
- 社区常见传染病防治追责
- 房地产运营管理:掌握市场动态
- 鲜花预订收条
- 国际旅行传染病防控培训制度
- 社区卫生服务-高血压评估指标
- 职业素养培训之如何提升职业素养课件
- 2023年版GMP附录《计算机化系统》解读
- 新浙教版初中数学教材完整目录
- 色彩原理与应用知到章节答案智慧树2023年上海出版印刷高等专科学校
- 四川省绵阳富乐国际2022-2023学年物理八下期中质量跟踪监视模拟试题含解析
- 机械设计手册第六版pdf
- 宜昌市金东方学校2023年小升初面试题(语数外)
- 型截面梁的整体稳定性实验报告
- 全国初中信息技术优质课一等奖《flash动画制作-引导线的应用》课件
- YY/T 0292.1-2020医用诊断X射线辐射防护器具第1部分:材料衰减性能的测定
- GB/T 37364.1-2019陆生野生动物及其栖息地调查技术规程第1部分:导则
评论
0/150
提交评论