Discrete Hilbert transformations can be implemented in either the time or frequency domains. Let's look at timedomain Hilbert transformers first.
9.4.1 TimeDomain Hilbert Transformation: FIR Filter Implementation
Looking back at Figure 94, and having h(n) available, we want to know how to generate the discrete xi(n). Recalling the frequencydomain product in Eq. (91), we can say xi(n) is the convolution of xr(n) and h(k). Mathematically, this is:
So this means we can implement a Hilbert transformer as a discrete nonrecursive finite impulse response (FIR) filter structure as shown in Figure 910.
Figure 910. FIR implementation of a Ktap Hilbert transformer.
Designing a traditional timedomain FIR Hilbert transformer amounts to determining those h(k) values so the functional block diagram in Figure 94 can be implemented. Our first thought is merely to take the h(n) coefficient values from Eq. (911), or Figure 99, and use them for the h(k)'s in Figure 910. That's almost the right answer. Unfortunately, the Figure 99 h(n) sequence is infinite in length, so we have to truncate the sequence. Figuring out what the truncated h(n) should be is where the true design activity takes place.
To start with, we have to decide if our truncated h(n) sequence will have an odd or even length. We make this decision by recalling that FIR implementations having antisymmetric coefficients and an odd, or even, number of taps are called a Type III, or a Type IV, system respectively[13]. These two antisymmetric filter types have the following unavoidable restrictions with respect to their frequency magnitude responses H(w):
h(n) length: 
Odd (Type III) 
Even (Type IV) 
H(0) = 0 
H(0) = 0 

H(ws/2) = 0 
H(ws/2) no restriction 
What this little table tells us is oddtap Hilbert transformers always have a zero magnitude response at both zero Hz and at half the sample rate. Eventap Hilbert transformers always have a zero magnitude response at zero Hz. Let's look at some examples.
Figure 911 shows the frequency response of a 15tap (Type III, oddtap) FIR Hilbert transformer whose coefficients are designated as h1(k). These plots have much to teach us.
Figure 911. H1(w) frequency response of h1(k), a 15tap Hilbert transformer.
In our relentless pursuit of correct results, we're forced to compensate for the linear phase shift of H1(w)—that constant time value equal to the group delay of the filter—when we generate our analytic xc(n). We do this by delaying, in time, the original xr(n) by an amount equal to the group delay of the h1(k) FIR Hilbert transformer. Recall that the group delay G of a Ktap FIR filter, measured in samples, is G = (K–1)/2 samples. So our block diagram for generating a complex xc(n) signal, using an FIR structure, is given in Figure 912(a). There we delay xr(n) by G = (7–1)/2 = 3 samples, generating the delayed sequence x'r(n). This delayed sequence now aligns properly in time with xi(n).
Figure 912. Generating an xc(n) sequence when h(k) is a 7tap FIR Hilbert filter: (a) processing steps; (b) filter structure.
If you're building your oddtap FIR Hilbert transform in hardware, an easy way to obtain x'r(n) is to tap off the original xr(n) sequence at the center tap of the FIR Hilbert transformer structure as in Figure 912(b). If you're modeling Figure 912(a) in software, the x'r(n) sequence can be had by inserting G = 3 zeros at the beginning of the original xr(n) sequence.
We can, for example, implement an FIR Hilbert transformer using a Type IV FIR structure, with its even number of taps. Figure 913 shows this notion where the coefficients are, say, h2(k). See how the frequency magnitude response is nonzero at ±fs/2 Hz. Thus this eventap filter approximates an ideal Hilbert transformer somewhat better than an oddtap implementation.
Figure 913. H2(w) frequency response of h2(k), a 14tap Hilbert transformer.
One of the problems with this traditional Hilbert transformer is that the passband gain in H2(w) is not unity for all frequencies, as is the x'r(n) path in Figure 912. So to minimize errors, we must use many h2(k) coefficients (or window the coefficients) to make H2(w)'s passband as flat as possible.
Although not shown here, the negative slope of the phase response of H2(w) corresponds to a filter group delay of G = (14–1)/2 = 6.5 samples. This requires us to delay the original xr(n) sequence by a noninteger (fractional) number of samples in order to achieve time alignment with xi(n). Fractional time delay filters is beyond the scope of this material, but Reference [4] is a source of further information on the topic.
Let's recall that alternate coefficients of a Type III (oddtap) FIR are zeros. Thus the oddtap Hilbert transformer is more attractive than an eventap version from a computational workload standpoint. Almost half of the multiplications in Figure 910 can be eliminated for a Type III FIR Hilbert transformer. Designers might even be able to further reduce the number of multiplications by a factor of two by using the folded FIR structure (discussed in Section 13.7) that's possible with symmetric coefficients (keeping in mind that half the coefficients are negative).
A brief warning: here's a mistake sometimes even the professionals make. When we design standard linearphase FIR filters, we calculate the coefficients and then use them in our hardware or software designs. Sometimes we forget to flip the coefficients before we use them in an FIR filter. This forgetfulness usually doesn't hurt us because typical FIR coefficients are symmetrical. Not so with FIR Hilbert filters, so please don't forget to reverse the order of your coefficients before you use them for convolutional filtering. Failing to flip the coefficients will distort the desired HT phase response.
As an aside, Hilbert transformers can be built with IIR filter structures, and in some cases they're more computationally efficient than FIR Hilbert transformers at the expense of a slight degradation in the 90o phase difference between xr(n) and xi(n)[5,6].
9.4.2 FrequencyDomain Hilbert Transformation
Here's a frequencydomain Hilbert processing scheme deserving mention because the HT of xr(n) and the analytic xc(n) sequence can be generated simultaneously. We merely take an Npoint DFT of a real evenlengthN xr(n) signal sequence, obtaining the discrete Xr(m) spectrum given in Figure 914(a). Next, create a new spectrum Xc(m) = 2Xr(m). Set the negativefrequency Xc(m) samples, that's (N/2)+1 m N–1, to zero leaving us with a new onesided Xc(m) spectrum as in Figure 914(b). Next divide the Xc(0) (the DC term) and the Xc(N/2) spectral samples by two. Finally, we perform an Npoint inverse DFT of the new Xc(m), the result being the desired analytic xc(n) timedomain sequence. The real part of xc(n) is the original xr(n), and the imaginary part of xc(n) is the HT of xr(n). Done!
Figure 914. Spectrum of original xr(n) sequence, and the onesided spectrum of analytic xc(n) sequence.
There are several issues to keep in mind concerning this straightforward frequencydomain analytic signal generation scheme:
Should you desire a decimatedbytwo analytic x'c(n) sequence based on xr(n), it's easy to do, thanks to Reference [7]. First, compute the Npoint Xr(m). Next, create a new spectral sequence X'c(k) = 2Xr(k) for 1 k (N/2)1. Set X'c(0) equal to Xr(0) + Xr(N/2). Finally, compute the (N/2)point inverse DFT of X'c(m) yielding the decimatedbytwo analytic x'c(n). The x'c(n) sequence has a sample rate of f's = fs/2, and the spectrum shown in Figure 914(c).
In Section 13.28.2 we discuss a scheme to generate interpolated analytic signals from xr(n).
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