![Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing
• We live in a continuous-time world: most of the
signals we encounter are CT signals, e.g. xc(t)
• How do we convert them into DT signals x[n]?
• Sampling, taking snap shots of xc(t) every T
seconds
• T is the sampling period
x[n] = xc(nT), n = ..., -1, 0, 1, 2, …
• regularly spaced samples
• How do we perform sampling?
Periodic Sampling](https://image.slidesharecdn.com/dsplecture7-231118061306-67b32680/85/DSP-Lecture-7-pdf-5-320.jpg)
![Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing
Sampling can be viewed as modulating with impulse
train
• xc(t) continuous time signal
• s(t) periodic impulse train with fundamental period T
• xs(t) sampled continuous version which is product of
xc(t) and s(t)
• x[n] discrete time signal
Periodic Sampling](https://image.slidesharecdn.com/dsplecture7-231118061306-67b32680/85/DSP-Lecture-7-pdf-7-320.jpg)
![Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing
xc(t)
s(t)
xs(t)
x[n]
Periodic Sampling](https://image.slidesharecdn.com/dsplecture7-231118061306-67b32680/85/DSP-Lecture-7-pdf-9-320.jpg)
DSP Lecture 7.pdf
- 1. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing DIGITAL SIGNAL PROCESSING
- 2. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Lecture 6: Summary • Inverse z-Transform • Properties of z-Transform • LTI System Characterization
- 3. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Lecture 7: Outline • Periodic Sampling • Frequency Domain Representation of Sampling • Reconstruction of a Bandlimited Signal From its Samples
- 4. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Analog to Digital Conversion and Sampling
- 5. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing • We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. xc(t) • How do we convert them into DT signals x[n]? • Sampling, taking snap shots of xc(t) every T seconds • T is the sampling period x[n] = xc(nT), n = ..., -1, 0, 1, 2, … • regularly spaced samples • How do we perform sampling? Periodic Sampling
- 6. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Sampling can be viewed as modulating with impulse train Periodic Sampling
- 7. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Sampling can be viewed as modulating with impulse train • xc(t) continuous time signal • s(t) periodic impulse train with fundamental period T • xs(t) sampled continuous version which is product of xc(t) and s(t) • x[n] discrete time signal Periodic Sampling
- 8. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Periodic Sampling Sampling can be viewed as modulating with impulse train
- 9. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing xc(t) s(t) xs(t) x[n] Periodic Sampling
- 10. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Frequency domain representation of Sampling • CTFT pair • DTFT pair Periodic Sampling
- 11. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Frequency domain representation of Sampling Periodic Sampling • By using multiplication property of CTFT • CTFT of periodic impulse train is
- 12. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Frequency domain representation of Sampling Periodic Sampling band-limited signal Xc(jΩ)=0 for |Ω|> ΩN Xs(jΩ) drawn assuming Ωs – ΩN > ΩN Ωs > 2ΩN
- 13. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Frequency domain representation of Sampling Periodic Sampling band-limited signal Xc(jΩ)=0 for |Ω|> ΩN Xs(jΩ) drawn assuming Ωs – ΩN < ΩN Ωs < 2ΩN
- 14. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of xc(t) from its Sampled Signal xs(t) •xc(t) can be recovered exactly from xs(t) by means of a lowpass filter with gain T and a cutoff frequency greater than ΩN and less than Ωs – ΩN, if Ωs > 2ΩN Periodic Sampling
- 15. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of xc(t) from its Sampled Signal xs(t) Periodic Sampling
- 16. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing The Sampling Theorem Suppose xc(t) is bandlimited, so that X(jΩ) = 0 for |Ω| > ΩN Then xc(t) is uniquely determined by its samples xc(nT), n = 0, ±1, ±2, ... , if Ωs > 2ΩN = The Nyquist rate Where Ωs = 2π/T Periodic Sampling
- 17. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Periodic Sampling • As we know • Apply CTFT on above equation • As
- 18. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Periodic Sampling • Therefore
- 19. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of Bandlimited Signal from its Samples • If Sampling Theorem is satisfied The original continuous-time signal can be recovered by filtering sampled signal with an ideal low-pass filter (LPF)
- 20. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Ideal Reconstruction Filter • Ideal LPC with cut of frequency of c=/T or fc=2/T T / t T / t sin t hr
- 21. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of Bandlimited Signal from its Samples • The reconstructed signal xr(t) can be written as
- 22. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of Bandlimited Signal from its Samples n r T / nT t T / nT t sin n x t x sinc function is 1 at t=0 sinc function is 0 at nT
- 23. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Reconstruction of Bandlimited Signal from its Samples • The reconstructed signal in frequency domain
- 24. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Oversampling
- 25. Muhammad Majid (m.majid@uettaxila.edu.pk) Digital Signal Processing Undersampling (Aliasing Distortion)