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4.4.3 Interpolation Using Unchanged Key Values

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4.4.3 Interpolation Using Unchanged Key Values

It is often necessary to retain the values from the input sequence y(m) in the interpolated x(n). without changing them. In this case,

x(mL) = y(m) (4.18)

These key values correspond to values of the polyphase components x0(p)(n) of the output signal x(n):

To satisfy the requirement in (4.20), the impulse response component go(m) must be the unit impulse sequence: g0(m) = (m) (4.21)

In general, we allow a delay of the interpolated output signal x(n) with respect to the input signal y(m). The condition for interpolation with unchanged key values is thusg0(m) = (m - m0) (4.22)A filter with the property in (4.22) is an Mth–band filter, see also section 2.6.6 and (2.87). As far as the various scalings in (4.22) and (2.87) are concerned, the reader is referred to (1.91) and Fig. 1.30b.

The linear interpolator also satisfies the condition in (4.22). Its impulse response is exactly equal to zero when n = 0 and n = 2L, and has the value 1 at n = L, see Fig 4.19. Between these values it increases and decreases linearly. It can be easily seen that with this interpolator, the interpolated values lie on the straight line between the key points.

4.4.4 Representation with Output Commutator

Fig. 4.20 shows a generalized polyphase interpolator, which consists of L branches, where L is the interpolation factor. The interleaved filter components g0(m), g1(m) .... gL-1(m) exactly rebuild the original anti-imaging filter g(n).

The process of combining the subsequences x0(m), x1(m) .... xL-1(m) to produce the final output signal x(n) is implemented by first generating the polyphase components of the output signal. These polyphase components x(p)(n) are produced from the signal components x (m) by upsampling by a factor L and adding a delay z-(at the higher sampling rate ), in accordance with (4.17) , for =0,1,2,…L-1, as shown in Fig. 4.20. Finally, the polyphase components are all added together to produce the output signal x(n). The delay z-in the polyphase branches is realized in a simpler form as a sequence of delays in the output, as shown in Fig. 4.21.

Some authors also use a commutator in the output to represent the combining of the signals x(m) into the output signal x(n) , as shown in Fig.4.22. The two representations are, in fact, equivalent.Fig. 4.23 shows the timing relationship between the output signal of the polyphase interpolator and the input signal. Each new sample value in the input sequence y(m) is simultaneously sent to all of the subfilters g(m). This allows the current values of the signal components x(m) to be calculated.

4.4.5 Memory-Saving polyphase Structures

As with polyphase decimators, the various delays of the subfilters in polyphase interpolators can be strung together, thus forming a single set of delays. To do this, the subfilters must be implemented n the first direct form (see Fig. 2.3.)

Fig. 4.24 shows the polyphase interpolator with the output commutator and subfilters in the first direct from. Since the input sequence is same in the each of the three delay branches, they can be brought together into a single chain, as in Fig.4.25.Instead of periodically switching between the subfilters, it is possible to periodically exchange the filter coefficients, as in Fig.4.26.

4.4.6 Representation of polyphase Interpolation Using Z-transforms

In what follows, z-transforms are used to derived a polyphase interpolator that is the dual of the structure in Fig. 4.16. This starting point is the generalized interpolation structure in Fig. 4.27.

y(z) v(z) x(z)

G(z)

Figure 4.28 Original form of a polyphase interpolator (a), and a more efficient and memory saving version(b)

a)

Y(z)

X(z)

G0(zL)

z-1

G1(zL)

z-1

G2(zL)

z-1

GL-1(zL)

b)

Y(z)

X(z)

G0(z)

z-1

G1(z)

z-1

G2(z)

z-1

GL-1(z)

We can use (1.10) and (1.11) to write the transfer function G(z) of the interpolation filter as

Because of (4.23), we can replace the generalized in Fig 4.27 with the polyphase structure in Fig. 4.28a.

By using the fourth and sixth identities, we can drive the variant that is shown in Fig. 4.28b.

4.5 Decimation and Interpolation with Rational Sampling Factors

In some application, the ratio of the sampling rates at the input and output is not an integer. In this case, both interpolation and decimation occur. The quotient L/M is in this case a rational number.

4.5.1 Introduction

In order to change the sampling rate by the rational factor L/M, an interpolation with the factor L is followed by a decimation with the factor M, as shown in Fig.4.29

Y(z)

X(z)

Y1(z)

X1(z)

G(z)

Fig. 4.30a shows the input spectrum |Y1(e j )| for the normalized frequency . The normalized sampling rate is 2. Upsampling by a factor L yields the spectrum |Y1(e j )|in Fig. 4.30b, which differs only in the scaling on x-axis. This followed in Fig. 4.30c by the magnitude frequency response |G(e j )| of the band limiting filter. The stop-band of this filter must begin at or before = /M if M > L, and =/L if L > M. The signal |X1(e j )| in Fig. 4.30d has been sufficiently band-limited. Downsampling of this produces the output signal |X(e j )|.

4.5.2 Highly Efficient Converter Structures

In the following, a highly efficient sampling-rate converter with a non-integral conversion factor is derived, as also described in [Hsi 87]. It is assumed here that L/M < 1. For the case L/M >1, the same derivation can be used, but with the dual of all the structures.

In the first step, the upsampler L and the band-limiting filter G(z) in Fig. 4.29 are replaced, using the system in Fig. 4.28. In addition, the downsampler M is moved into each polyphase branch, in compliance with the first identity in Fig. 1.23. This gives the polyphase configuration in Fig. 4.31.

It can be seen in Fig. 4.32 that in all cases, each polyphase branch contains an FIR filter with the transfer function G(z), = 0,1,2 ... L-1, which is followed by a downsampler. In Fig. 4.32c, this is enclosed in a dotted box. A comparison with the Figs. 4.15 and 4.16b shows that each of the L polyphase components G(z) can further split into M polyphase components G μ(z), μ = 0,1,2 ... M-1. The result of this is shown in Fig. 4.33. With this solution, the filter operations can be performed at the lowest possible sampling rate.

4.5.3 Changing the sampling Rate by a factor 2/3

In the following example, the sampling rate is changed by a factor L/M= 2/3. The reader is also referred to [Vai 90]. This conversion is used for mixing consoles in audio studios, where the sampling rates of audio signal originally sampled at 32 kHz and 48 kHz need to be interchanged. Thus the following section deals with the conversion 48kHz 32kHz.

Fig 4.34a shows the general structure of the sampling rate converter with the conversion factor (the ratio of the sampling rate at the output to that of the input) L/M =2/3. The band-limiting filter G(z) works with twice the input sampling rate. The stop-band cutoff frequency, referred to this rate, is s =/3.In a first step, the interpolator is replaced by a polyphase structure in accordance with Fig. 4.28b. This is shown in fig. 4.34b. the delay z-1 is then replaced by the product z2 z-3 , Fig4.34c.

The factor z2 is moved through the upsampler L=2, and the factor z-3 through the downsampler M=3, Fig. 4.34d. In the last step, the order of the up and downsamplers is changed, and the non-causal operation z1 replaced by additional delay z-1 at the input, Fig.4.34e.

If we replace the decimators in the two polyphase branches in Fig. 4.34e with polyphase decimators, in accordance with Fig. 4.16b, we obtain the final structure of Fig. 4.35. In the mixing console mentioned above, the filter operations in the original structure in the Fig. 4.34a are operated at a sampling rate of 16kHz, while the same operations in the original structure in the Fig. 4.34a are operated at 96 kHz. The actual number of filter operations is reduced by the factor L·M= 6.

4.5.4 Dual Sampling-Rate Converter

In order to get sampling rate conversion with the factor 3/2 for example from 32 kHz to 48 kHz, a band limiting filter with stop- band frequency s =/3 is likewise used. The duals of the steps in Fig. 4.34 need to be performed.

The result of this alteration is a structure that is the dual of the one in Fig. 4.35 with the same computational efficiency.