1 / 64

Outline of chapter seven

Outline of chapter seven. Discrete-Time Fourier Transform Discrete Fourier Transform Properties of the DFT System Analysis via the DTFT and DFT FFT Algorithm Applications of the FFT Algorithm. Discrete-Time Fourier Transform (DTFT) （离散时间傅里叶变换）.

reevescharles
Download Presentation

Outline of chapter seven

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Outline of chapter seven • Discrete-Time Fourier Transform • Discrete Fourier Transform • Properties of the DFT • System Analysis via the DTFT and DFT • FFT Algorithm • Applications of the FFT Algorithm

  2. Discrete-Time Fourier Transform (DTFT)（离散时间傅里叶变换） DTFT transforms a discrete signal x(n) into in general a complex- valued continuous function of the real variable , calleddigital frequency（数字频率）, which is measured in radians（弧度）. DTFT： Existence Condition: x[n] is absolutely summable,that is

  3. 数字频率 与模拟频率 之间的对应关系

  4. DTFT is a periodic function with period

  5. Spectrum Frequency spectrum of : Polar form （极坐标形式）: Amplitude spectrum （幅度谱）of : Phase spectrum （相位谱）of :

  6. Inverse DTFT（离散时间傅里叶反变换） IDTFT（反变换)： Due to the periodicity of , the integral can be evaluated over any interval of length

  7. Example 7.2 Consider the discrete-time signal

  8. Example 7.2 (con’t)

  9. Example 7.3（矩形脉冲的DTFT)

  10. An important conclusion Recall that is a periodic function of with period Low frequency: High frequency:

  11. is real valued is an even function , is an odd function, that is and Proof: as a result and

  12. Signals with Even and Odd symmetry If the real-valued discrete-time signal is an even function If the real-valued discrete-time signal is an odd function

  13. Properties of the DTFT • Linearity • Right or left shift in time • Time reversal • Multiplication by n • Multiplication by a complex exponential • Convolution in the time domain • Parseval’s theorem P309

  14. Transform pairs （变换对）

  15. Discrete Fourier Transform (DFT)（离散傅里叶变换） To implement DTFT techniques on a digital computer, it is necessary to discretize in frequency. This leads to the concept of the discrete Fourier transform. Definition: N-point DFT Definition: N-point Inverse DFT(IDFT)

  16. Example 7.8 Computation of DFT

  17. Example 7.9 Computation of inverse DFT

  18. Relationship to DTFT Given a discrete-time signal with , for and Comparing the two equations reveals that Thus the DFT can be view as a frequency-sampled version of the DTFT

  19. Example 7.10

  20. Example 7.10

  21. Example 7.10 (con’t)

  22. Example 7.12

  23. Example 7.12 (cont.)

  24. Example 7.12 (cont.)

  25. 取模运算 如果 ， m为整数；则有： 此运算符表示n被N除，商为m，余数为 。 是 的解,或称作取余数,或说作n对N取模值, 或简称为取模值,n模N。

  26. 取模运算示例

  27. Circular Time Shift (圆周移位） 周期延拓

  28. Circular Time Shift

  29. Time reversal (时间反转）

  30. Linear Convolution (线性卷积） If both and are zero, for and is zero for and

  31. Circular Convolution (圆周卷积） Because N-point DFT of will not incorporate the values for , it is necessary to define a convolution operation so that the convolved signal is zero outside the range

  32. Example of linear convolution and circular convolution

  33. The relationship between linear circular convolution and circular convolution Setting , an L-step interval linear convolution and circular convolution give the same result, that is

  34. Properties of the DFT • Linearity • Circular time shift • Time reversal • Multiplication by a complex exponential • Circular convolution • Multiplication in the time domain • Parseval’s theorem P331

  35. System analysis via the DTFT and DFT The (N+Q)-point DFT of is The (N+Q)-point DFT of is given by Thus (N+Q)-point DFT of is given by

  36. System analysis via the DTFT and DFT (cont.) 称为系统的频率响应函数 Consider a linear time-invariant discrete-time system with output response Thus the DTFT of the output response is given by

  37. Response to a sinusoidal input Suppose that the input to the LTI discrete-time system with unit-pulse response is the sinusoid The output of the system is given by

  38. 例7.1 4 对余弦信号的响应

  39. 例7.1 4 对余弦信号的响应

  40. 例7.1 4 对余弦信号的响应 http://users.ece.gatech.edu/~bonnie/book Errata // 勘误表

  41. Fast Fourier Transform (FFT)（快速傅里叶变换） DFT的定义 Notation 注意：FFT并不是一种新的变换，它是DFT 的快速算法

  42. Fast Fourier Transform (FFT)（快速傅里叶变换） Notation （性质）Properties of 对称性 周期性 可约性 可约性

  43. FFT algorithm The length N of DFT must satisfy Divide the data into two group, that is Compute the DFT of x[n] for

  44. FFT algorithm (cont.)

  45. FFT (cont.) Consider that Butterfly computation (蝶型运算)

  46. FFT (cont.) N点DFT的第一次时域抽取分解图(N=8)

  47. FFT (cont.) N点DFT的第二次时域抽取分解图(N=8)

  48. FFT (cont.) N点FFT第三次时域抽取分解图(N=8)

More Related