44
Practical MATLAB
® Applications for Engineers
R.1.127 Reconstruction of an analog signal f(t) from the discrete sequence f(n) can be
accomplished, if the analog signal f(t) is band-limited and converted into the discrete sequence f(n) by sampling it with a rate above the Nyquist/Shannon rate.
R.1.128 Recall that the sampling theorem (Nyquist/Shannon) states that an analog signal
f(t), band-limited to fm hertz, must be sampled at a rate greater than twice its highest frequency fm, to be able to reconstruct f(t) from its samples given by f(n) (R.1.1).
The reconstruction process is done by passing the sequence f(n) through an ideal
analog low-pass fi lter with a cutoff frequency of fm (see Chapters 4 and 6 for information regarding frequency domain and fi ltering).
R.1.129 Since ideal sharp rectangular fi lters, with a cutoff frequency of fm are unrealizable, the practical sampling rate is often fi ve or six times the frequency fm. At this
point let us analyze the reconstruction process in the time domain, where the
reconstructed signal f(t) can be approximated by the following equation:
f t
f n
c t nT
T
n
n
( )
( )
sin (
)
ϭ
ϭ ϱ
ϭϩϱ
Ϫ
Ϫ
∑
where
sin ( )
sin( )
c t
t
t
ϭ
R.1.130 If a discrete time signal f(n) was originally obtained by sampling a band-limited
continuous time signal by using the Nyquist/Shannon sampling rate, then the
process of up- or down-sampling the time continuous signal f(t) means that the
original signal is sampled now by a higher or lower sampling rate.
R.1.131 The term up-sampling refers to increasing the sampling rate by an integer factor
L. This process means that interpolation is required by placing additional samples
between the original sampled function f(n). This process is accomplished by a lowpass fi lter scaled to a cutoff frequency w c = π/L.
R.1.132 The MATLAB function interp returns the sequence fL that consists of increasing
the sequence f(n) defi ned by a vector f by an integer factor L.
The function interp can take any of the following forms:
a. fL = interp(f, L)
b. fL = interp(f, LN, a)
c. [fL, h] = interp(f, L,N, a) (with default values of L = 4 and a = 0.5)
The resampled vector fL has a length given by length(fL) = L * length(f).
The sample signal f is assumed to be band-limited scaled to 0 ≤ w ≤ a, with a ≤ 0.5,
where h represents the interpolation fi lter coeffi cient. Ideally L ≤ 10.
R.1.133 The term decimate is referred to the reduction or down-sampling the discrete signal f by an integer factor M, returning the sequence fM.
The MATLAB function decimate can take any of the following forms:
fM = decimate(f, M)
fM = decimate(f, M . n)
fM = decimate(f, M . ’fi r’)
fM = decimate(f, M, N, ’fi r’)
CRC_47760_CH001.indd 44
CRC_47760_CH001.indd 44
7/25/2008 4:16:18 PM
7/25/2008 4:16:18 PM
Practical MATLAB
® Applications for Engineers
R.1.127 Reconstruction of an analog signal f(t) from the discrete sequence f(n) can be
accomplished, if the analog signal f(t) is band-limited and converted into the discrete sequence f(n) by sampling it with a rate above the Nyquist/Shannon rate.
R.1.128 Recall that the sampling theorem (Nyquist/Shannon) states that an analog signal
f(t), band-limited to fm hertz, must be sampled at a rate greater than twice its highest frequency fm, to be able to reconstruct f(t) from its samples given by f(n) (R.1.1).
The reconstruction process is done by passing the sequence f(n) through an ideal
analog low-pass fi lter with a cutoff frequency of fm (see Chapters 4 and 6 for information regarding frequency domain and fi ltering).
R.1.129 Since ideal sharp rectangular fi lters, with a cutoff frequency of fm are unrealizable, the practical sampling rate is often fi ve or six times the frequency fm. At this
point let us analyze the reconstruction process in the time domain, where the
reconstructed signal f(t) can be approximated by the following equation:
f t
f n
c t nT
T
n
n
( )
( )
sin (
)
ϭ
ϭ ϱ
ϭϩϱ
Ϫ
Ϫ
∑
where
sin ( )
sin( )
c t
t
t
ϭ
R.1.130 If a discrete time signal f(n) was originally obtained by sampling a band-limited
continuous time signal by using the Nyquist/Shannon sampling rate, then the
process of up- or down-sampling the time continuous signal f(t) means that the
original signal is sampled now by a higher or lower sampling rate.
R.1.131 The term up-sampling refers to increasing the sampling rate by an integer factor
L. This process means that interpolation is required by placing additional samples
between the original sampled function f(n). This process is accomplished by a lowpass fi lter scaled to a cutoff frequency w c = π/L.
R.1.132 The MATLAB function interp returns the sequence fL that consists of increasing
the sequence f(n) defi ned by a vector f by an integer factor L.
The function interp can take any of the following forms:
a. fL = interp(f, L)
b. fL = interp(f, LN, a)
c. [fL, h] = interp(f, L,N, a) (with default values of L = 4 and a = 0.5)
The resampled vector fL has a length given by length(fL) = L * length(f).
The sample signal f is assumed to be band-limited scaled to 0 ≤ w ≤ a, with a ≤ 0.5,
where h represents the interpolation fi lter coeffi cient. Ideally L ≤ 10.
R.1.133 The term decimate is referred to the reduction or down-sampling the discrete signal f by an integer factor M, returning the sequence fM.
The MATLAB function decimate can take any of the following forms:
fM = decimate(f, M)
fM = decimate(f, M . n)
fM = decimate(f, M . ’fi r’)
fM = decimate(f, M, N, ’fi r’)
CRC_47760_CH001.indd 44
CRC_47760_CH001.indd 44
7/25/2008 4:16:18 PM
7/25/2008 4:16:18 PM