1 / 16

Chapter 5 Ideal Filters, Sampling, and Reconstruction Sections 5.3-5.4 Wed. June 26, 2013

Chapter 5 Ideal Filters, Sampling, and Reconstruction Sections 5.3-5.4 Wed. June 26, 2013. 5.3.1 Ideal Filters – Linear Phase. To avoid phase distortion:. for all ω in the passband , where t d is a fixed positive number. Then the response to a sinusoidal input is:.

aden
Télécharger la présentation

Chapter 5 Ideal Filters, Sampling, and Reconstruction Sections 5.3-5.4 Wed. June 26, 2013

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. Chapter 5Ideal Filters, Sampling, and ReconstructionSections 5.3-5.4Wed. June 26, 2013

  2. 5.3.1 Ideal Filters – Linear Phase • To avoid phase distortion: • for all ω in the passband, where td is a fixed positive number. Then the response to a sinusoidal input is:

  3. 5.3.1 Ideal Filters – Linear Phase

  4. 5.3.2 Linear Phase Low-Pass Filter

  5. 5.3.2 Linear Phase Low-Pass Filter

  6. 5.3.2 Linear Phase Low-Pass Filter • Let’s do this in the time domain: • and using time-shift property:

  7. 5.3.2 Linear Phase Low-Pass Filter Impulse response starts before t = 0 Non-Causal !

  8. 5.3.2 Linear Phase Low-Pass Filter • Ideal filters are non-causal, due to the unrealistic vertical cutoffs. • Design of realistic causal filters will be addressed later.

  9. 5.4 Sampling where T is the sampling interval, and n is the sample number.

  10. 5.4 Sampling where T is the sampling interval, and n is the sample number.

  11. 5.4 Sampling As shown in text equations 5.49 – 5.51, the Fourier transform of the sampled signal is:

  12. 5.4 Sampling Original signal: Sampled signal:

  13. 5.4.1 Reconstruction Original signal: Sampled signal: Apply ideal lowpass filter with bandwidth between B and ωs-B.

  14. 5.4.1 Reconstruction Sampling theorem: Original signal can be recovered perfectly as long as it is sampled above the Nyquist rate. The Nyquist rate is twice the bandwidth of the original signal: In practice, signals are oversampled, i.e.,

  15. 5.4.2 Reconstruction Interpolation (reconstruction) formula is derived from the impulse response of the ideal low-pass filter: This shows how to “connect the dots” between the samples to recover the original signal. Notice that every sample contributes to every time point. It is not as simple as connecting the dots with straight lines!

  16. 5.4.3 Aliasing If the sampling rate is below the Nyquist rate, aliasing occurs.

More Related