Sampling Theorems for Random Records

A sample random time history record x(t) from a random process {x_k(t)} is considered and exists only for the time interval from 0 to T seconds, and is zero elsewhere. Its Fourier transform can be expressed as:

X(ω) = Z T

0

x(t)e^−j2πωtdt (3.5.3)

If x(t) is continually repeated, a periodic time function with a period of T seconds is obtained and the fundamental frequency increments is f = 1/T . By applying a Fourier series expansion, the x(t) can be re-written as:

x(t) =

∞

X

−∞

A_ne^j2πnt/T (3.5.4)

where A_n is computed from x(t) as:

A_n= 1 T

Z T 0

x(t)e^j2πnt/Tdt (3.5.5)

Thus, X(n/T ) determines An and, therefore, x(t) at all t:

Xn This result determines X(ω) for all ω, and it is the sampling theorem in the fre-quency domain. The fundamental frefre-quency increment 1/T is called a Nyquist co-interval.

In a similar way, a Fourier transform X(ω) of some sample random time his-tory record x(t) is considered. X(ω) exists only over a frequency interval from

−B to B H and is zero at all other frequencies. The actual realizable frequency band ranges from 0 to B Hz. The inverse Fourier transform leads to:

x(t) = Z B

−B

X(ω)e^j2πωtdω (3.5.7)

X(ω) is considered to be repeated in frequency in order to obtain a periodic frequency function with a period of 2B Hz, and the fundamental time incre-ment is t = 1/(2B). The Fourier transform X(ω) can be written as a Fourier series:

where Cn can be computed from X(ω) as:

C_n= 1 Thus, this determines x(t) for all t:

x(t) =

The last equation shows how x(t) is reconstructed from the sample values taken 1/(2B) seconds apart. This result is the sampling theorem in the time domain and the fundamental time increment 1/(2B) is called a Nyquist interval.

By sampling X(ω) at Nyquist co-interval points 1/T apart on the frequency scale from −B to B, the number of discrete samples required to describe x(t) is:

N = 2B

1/T = 2BT (3.5.12)

Whilst by sampling x(t) at Nyquist interval points 1/2B apart on the timescale from 0 to T it follows that:

N = T

1/2B = 2BT (3.5.13)

Sampling rates and aliasing errors

The sampling of analog signals for digital data analysis is usually performed at equally spaced time intervals. It is necessary to determine the appropriate

sampling interval ∆t. As discussed earlier, the minimum number of discrete samples required to describe the analog signal is given by N = 2BT . Therefore the maximum sampling interval is:

∆t = 1

2B (3.5.14)

Sampling at more closely spaced than _2B¹ points will yield correlated and re-dundant sample values, whilst sampling at points further apart than _2B¹ will lead to confusion between the low and high frequency components in the orig-inal data. This latter problem is called aliasing and is a source of error for the

Figure 3.9: Frequency aliasing due to an inadequate digital sampling rate.

processing of the data after the analog-digital conversion. The presence of high frequencies in the original signal could be misinterpreted in the discretisation process, and those frequencies will appear as low frequencies.

If the sampling frequency is f_s, then the signal of frequency ω and signal of frequency f_s− ω are indistinguishable after discretization, and this causes dis-tortion of the measured spectra using DFT. The highest frequency that can be defined by a sampling rate of 1/∆t is 1/(2∆t) Hz. Frequencies in the original data above 1/(2∆t) Hz will appear below 1/(2∆t) Hz and be confused with the data in this lower frequency range.

f_A= 1

2∆t (3.5.15)

is called the N yquistf requency.

The signal appears in DFT as a distorted form: towards the upper end of the applicable frequency range (0 − fA) the distortion is due to the fact that the

Figure 3.10: Aliasing error in the computation of an autospectral density function.

portion of the signal with frequency components above fA will be reflected in the range 0 − fA. The problem is solved using an anti-aliasing filter which subjects the original time signal to a low-pass, sharp cut-off filter. Actually, because the filters used are not perfect and have a finite cut-off rate, the spectral measurements in a frequency range approaching the Nyquist frequency must be rejected. Typically, the range from 0, 8 · f_A to f_A is rejected.

Leakage

Leakage is a direct consequence of the need to take only a finite length of time history coupled with the assumption of periodicity. We consider a sinusoidal signal. If the signal is perfectly periodic in the time window T , the resulting spectrum will be simply a single line at the frequency of the sine wave. If,

Figure 3.11: Leakage error.

however, the periodicity assumption is not satisfied and there is a discontinuity

at the end of the sample, the spectrum will not indicate the single frequency which the original time signal possessed, and this frequency may not even prevail in the spectral lines, because the energy leaks into a number of spectral lines close to the true frequency and the spectrum is spread over several lines.

Leakage is more relevant for low frequency signals and is a serious problem in may applications of digital signal processing, including FRF measurements.

There are several ways of minimizing its effects [7]:

• Changing the duration of the measurement sample length to match any underlying periodicity in the signal. This is possible only if the signal is periodic and its period can be determined.

• Increasing the duration of measurement time T, so the separation be-tween the spectral lines is finer.

• Windowing: consists on modifying the signal sample obtained in such a way as to reduce the severity of the leakage effect. This process is

Figure 3.12: Influence of Hanning window to Fourier transform of a signal.

the most widely employed in modal testing. A prescribed profile w(t) is imposed on the time signal prior to performing Fourier transform. The analyzed signal is then product of original signal and window profile as shown in the figure.