Presentation on theme: "HILBERT TRANSFORM Fourier, Laplace, and z-transforms change from the time-domain representation of a signal to the frequency-domain representation of the."— Presentation transcript:
1HILBERT TRANSFORMFourier, Laplace, and z-transforms change from the time-domain representation of a signal to the frequency-domain representation of the signalThe resulting two signals are equivalent representations of the same signal in terms of time or frequencyIn contrast, The Hilbert transform does not involve a change of domain, unlike many other transforms
2HILBERT TRANSFORMStrictly speaking, the Hilbert transform is not a transform in this senseFirst, the result of a Hilbert transform is not equivalent to the original signal, rather it is a completely different signalSecond, the Hilbert transform does not involve a domain change, i.e., the Hilbert transform of a signal x(t) is another signal denoted by in the same domain (i.e.,time domain)
3HILBERT TRANSFORMThe Hilbert transform of a signal x(t) is a signal whose frequency components lag the frequency components of x(t) by 90has exactly the same frequency components present in x(t) with the same amplitude–except there is a 90 phase delayThe Hilbert transform of x(t) = Acos(2f0t + ) is Acos(2f0t + - 90) = Asin(2f0t + )
4HILBERT TRANSFORM A delay of /2 at all frequencies ej2f0t will becomee-j2f0t will becomeAt positive frequencies, the spectrum of the signal is multiplied by -jAt negative frequencies, it is multiplied by +jThis is equivalent to saying that the spectrum (Fourier transform) of the signal is multiplied by-jsgn(f).
5HILBERT TRANSFORMAssume that x(t) is real and has no DC component : X(f)|f=0 = 0,thenThe operation of the Hilbert transform is equivalent to a convolution, i.e., filtering
6Example 2.6.1Determine the Hilbert transform of the signal x(t) = 2sinc(2t)SolutionWe use the frequency-domain approach . Using the scaling property of the Fourier transform, we haveIn this expression, the first term contains all the negative frequencies and the second term contains all the positive frequenciesTo obtain the frequency-domain representation of the Hilbert transform of x(t), we use the relation = -jsgn(f)F[x(t)], which results inTaking the inverse Fourier transform, we have
7HILBERT TRANSFORMObviously performing the Hilbert transform on a signal is equivalent to a 90 phase shift in all its frequency componentsTherefore, the only change that the Hilbert transform performs on a signal is changing its phaseThe amplitude of the frequency components of the signal do not change by performing the Hilbert-transformOn the other hand, since performing the Hilbert transform changes cosines into sines, the Hilbert transform of a signal x(t) is orthogonal to x(t)Also, since the Hilbert transform introduces a 90 phase shift, carrying it out twice causes a 180 phase shift, which can cause a sign reversal of the original signal
8HILBERT TRANSFORM - ITS PROPERTIES Evenness and OddnessThe Hilbert transform of an even signal is odd, and the Hilbert transform of an odd signal is evenProofIf x(t) is even, then X(f) is a real and even functionTherefore, -jsgn(f)X(f) is an imaginary and odd functionHence, its inverse Fourier transform will be oddIf x(t) is odd, then X(f) is imaginary and oddThus -jsgn(f)X(f) is real and evenTherefore, is even
9HILBERT TRANSFORM - ITS PROPERTIES Sign ReversalApplying the Hilbert-transform operation to a signal twice causes a sign reversal of the signal, i.e.,ProofX( f ) does not contain any impulses at the origin
10HILBERT TRANSFORM - ITS PROPERTIES EnergyThe energy content of a signal is equal to the energy content of its Hilbert transformProofUsing Rayleigh's theorem of the Fourier transform,Using the fact that |-jsgn(f)|2 = 1 except for f = 0, and the fact that X(f) does not contain any impulses at the origin completes the proof
11HILBERT TRANSFORM - ITS PROPERTIES OrthogonalityThe signal x(t) and its Hilbert transform are orthogonalProofUsing Parseval's theorem of the Fourier transform, we obtainIn the last step, we have used the fact that X(f) is Hermitian; | X(f)|2 is even
12Ideal filters Ideal lowpass filter An LTI system that can pass all frequencies less than some W and rejects all frequencies beyond WW is the bandwidth of the filter.
13Ideal filters Ideal highpass filter There is unity outside the interval -W f W and zero insideIdeal bandpass filterhave a frequency response that is unity in some interval–W1 |f| W2 and zero otherwiseThe bandwidth of the filter is W2 – W1
14Nonideal filters – 3dB bandwidth For nonideal filters, the bandwidth is usually defined as the band of frequencies at which the power of the filter is at least half of the maximum powerThis bandwidth is usually called the 3 dB bandwidth of the filter, because reducing the power by a factor of two is equivalent to decreasing it by 3 dB on the logarithmic scale
15LOWPASS AND BANDPASS SIGNALS Lowpass signalA signal in which the spectrum (frequency content) of the signal is located around the zero frequencyBandpass signalA signal with a spectrum far from the zero frequencyThe frequency spectrum of a bandpass signal is usually located around a frequency fcfc is much higher than the bandwidth of the signal