(b)If f is real and causal, and f(O) is finite, then R(k) and X(k) are related by the Hilbert transforms X() R(k) dk, R(=-Pv X(k) (c)If f is causal and has finite energy, it is not possible to have F(k)=0 for k I < ks k?. That is. the transform of a causal function cannot vanish over an interval A causal function is completely determined by the real or imaginary part of its spectrum. As with item 4, this is helpful when performing calculations or mea- surements in the frequency domain. If the function is not band-limited however truncation of integrals will give erroneous results 7. Time-limited us. band-limited functions. Assume t2>t1. If f()=0 for both t < tr and t>I2, then it is not possible to have F()=0 for both k kI and k > k2 where k2>k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier tegral theorem is useful when solving homogeneous partial differential equations n the frequency domain. 9. Space or time shift. For any fixed xo f(x-xo)+ F(k)- Jxta. a temporal or spatial shift affects only the phase of the transform, not the magni- tude 10. Frequency shift. For any fixed ko f(r)e Note that if f < F where f is real, then frequency-shifting F causes f to be- come complex -again, this is important if F has been obtained experimentally through computation in the fre 11. Similarity. We have where a is any real constant. " Reciprocal spreading" is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and 12. Convolution. We have fi(rf2(x-x)+ Fi(k) F2(k) f1(x)f2(x)分 Fc)F2(k-k)di 0 2001 by CRC Press LLC(b) If f is real and causal, and f (0) is finite, then R(k) and X(k) are related by the Hilbert transforms X(k) = − 1 π P.V. ∞ −∞ R(k) k − k dk , R(k) = 1 π P.V. ∞ −∞ X(k) k − k dk . (c) If f is causal and has finite energy, it is not possible to have F(k) = 0 for k1 < k < k2. That is, the transform of a causal function cannot vanish over an interval. A causal function is completely determined by the real or imaginary part of its spectrum. As with item 4, this is helpful when performing calculations or mea￾surements in the frequency domain. If the function is not band-limited however, truncation of integrals will give erroneous results. 7. Time-limited vs. band-limited functions. Assume t2 > t1. If f (t) = 0 for both t < t1 and t > t2, then it is not possible to have F(k) = 0 for both k < k1 and k > k2 where k2 > k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited. 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier integral theorem is useful when solving homogeneous partial differential equations in the frequency domain. 9. Space or time shift. For any fixed x0, f (x − x0) ↔ F(k)e− jkx0 . (A.3) A temporal or spatial shift affects only the phase of the transform, not the magni￾tude. 10. Frequency shift. For any fixed k0, f (x)e jk0 x ↔ F(k − k0). Note that if f ↔ F where f is real, then frequency-shifting F causes f to be￾come complex — again, this is important if F has been obtained experimentally or through computation in the frequency domain. 11. Similarity. We have f (αx) ↔ 1 |α| F  k α  , where α is any real constant. “Reciprocal spreading” is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and vice versa. 12. Convolution. We have ∞ −∞ f1(x ) f2(x − x ) dx ↔ F1(k)F2(k) and f1(x) f2(x) ↔ 1 2π ∞ −∞ F1(k )F2(k − k ) dk .