40
D.-W. Kim and C.-H. Im
It is important to understand that the sampling frequency f s is the only factor that
determines the Nyquist frequency.
As in the CFT, the Fourier coefficients X[k] are complex numbers. Therefore,
X[k] can be expressed as the following in Cartesian form:
X [k] X Re [k] + jX Im [k],
(3.7)
where X Re [k] and X Im [k] are the real and imaginary parts of X [k], respectively. From
(3.7), the magnitude |X [k]| and the phase Φ[k] can be expressed as
|X [k]|
X
2
Re [k] + X
2
Im [k],
(3.8)
Φ[k] tan
−1 X Im [k]
X Re [k]
.
(3.9)
The inverse DFT transforms the Fourier coefficient X[k] back into the discrete
time series x[n], as follows:
x[n]
1
N
N −1
n0
X [k]e
j2πkn/N
.
(3.10)
3.2.3 Fast Fourier Transform
The fast Fourier transform (FFT) is a particular implementation of the DFT that gives
identical results with reduced calculations [10]. The calculation of (3.4) requires N
2
complex multiplications, because, for each of the N discrete frequencies, it is necessary to calculate the sum of N multiplications of complex exponentials. However,
in cases when N is a power of 2 (e.g., 64, 128, 512, 1024, …), many of these multiplications result in identical values, and many of the complex exponentials are zero
or 1. When redundant computations are removed, the number of multiplications is
reduced to Nlog 2 (N) rather than N
2 , reducing the computational burden, especially
when N is large. Spectral estimation based on the FFT assumes that the signal is
stationary and slowly varying.
3.2.4 Aliasing and Leakage
Spectral estimation based on the FFT has intrinsic properties called aliasing and
leakage [28], which need to be considered carefully. To understand the aliasing, consider a continuous signal with a single frequency, as shown in Fig. 3.2. As the signal
is recorded, the original signal is sampled to a discrete-time signal, depending on
the sampling frequency (or interval) of the analog-to-digital converter (ADC). If the
D.-W. Kim and C.-H. Im
It is important to understand that the sampling frequency f s is the only factor that
determines the Nyquist frequency.
As in the CFT, the Fourier coefficients X[k] are complex numbers. Therefore,
X[k] can be expressed as the following in Cartesian form:
X [k] X Re [k] + jX Im [k],
(3.7)
where X Re [k] and X Im [k] are the real and imaginary parts of X [k], respectively. From
(3.7), the magnitude |X [k]| and the phase Φ[k] can be expressed as
|X [k]|
X
2
Re [k] + X
2
Im [k],
(3.8)
Φ[k] tan
−1 X Im [k]
X Re [k]
.
(3.9)
The inverse DFT transforms the Fourier coefficient X[k] back into the discrete
time series x[n], as follows:
x[n]
1
N
N −1
n0
X [k]e
j2πkn/N
.
(3.10)
3.2.3 Fast Fourier Transform
The fast Fourier transform (FFT) is a particular implementation of the DFT that gives
identical results with reduced calculations [10]. The calculation of (3.4) requires N
2
complex multiplications, because, for each of the N discrete frequencies, it is necessary to calculate the sum of N multiplications of complex exponentials. However,
in cases when N is a power of 2 (e.g., 64, 128, 512, 1024, …), many of these multiplications result in identical values, and many of the complex exponentials are zero
or 1. When redundant computations are removed, the number of multiplications is
reduced to Nlog 2 (N) rather than N
2 , reducing the computational burden, especially
when N is large. Spectral estimation based on the FFT assumes that the signal is
stationary and slowly varying.
3.2.4 Aliasing and Leakage
Spectral estimation based on the FFT has intrinsic properties called aliasing and
leakage [28], which need to be considered carefully. To understand the aliasing, consider a continuous signal with a single frequency, as shown in Fig. 3.2. As the signal
is recorded, the original signal is sampled to a discrete-time signal, depending on
the sampling frequency (or interval) of the analog-to-digital converter (ADC). If the