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
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
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
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
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
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
