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Multirate filters alter the sample rate of the input signal during the filtering process. Such filters are useful in both rate conversion and filter bank applications.

The Dyadic Analysis Filter Bank block decomposes a broadband signal into a collection of subbands with smaller bandwidths and slower sample rates. The Dyadic Synthesis Filter Bank block reconstructs a signal decomposed by the Dyadic Analysis Filter Bank block.

To use a dyadic synthesis filter bank to perfectly reconstruct
the output of a dyadic analysis filter bank, the number of levels
and tree structures of both filter banks *must* be
the same. In addition, the filters in the synthesis filter bank *must* be
designed to perfectly reconstruct the outputs of the analysis filter
bank. Otherwise, the reconstruction will not be perfect.

Dyadic analysis filter banks are constructed from the following basic unit. The unit can be cascaded to construct dyadic analysis filter banks with either a symmetric or asymmetric tree structure.

Each unit consists of a lowpass (LP) and highpass (HP) FIR filter
pair, followed by a decimation by a factor of 2. The filters are halfband
filters with a cutoff frequency of *F _{s}* / 4, a quarter of the input sampling
frequency. Each filter passes the frequency band that the other filter
stops.

The unit decomposes its input into adjacent high-frequency and low-frequency subbands. Compared to the input, each subband has half the bandwidth (due to the half-band filters) and half the sample rate (due to the decimation by 2).

The following figures illustrate the *concept* of
a filter bank, but *not* how the block implements
a filter bank; the block uses a more efficient polyphase implementation.

**n-Level Asymmetric Dyadic Analysis Filter Bank**

Use the above figure and the following figure to compare the two tree structures of the dyadic analysis filter bank. Note that the asymmetric structure decomposes only the low-frequency output from each level, while the symmetric structure decomposes the high- and low-frequency subbands output from each level.

**n-Level Symmetric Dyadic Analysis Filter Bank**

The following table summarizes the key characteristics of the symmetric and asymmetric dyadic analysis filter bank.

**Notable Characteristics of Asymmetric and Symmetric
Dyadic Analysis Filter Banks**

Characteristic | N-Level Symmetric | N-Level Asymmetric |
---|---|---|

| All the low-frequency and high-frequency subbands in a level are decomposed in the next level. | Each level's low-frequency subband is decomposed in the next level, and each level's high-frequency band is an output of the filter bank. |

| 2 | n+1 |

| For an input with bandwidth N / 2 samples.^{n} | For an input with bandwidth BW and N samples, BW, and _{k}N samples,
where_{k}$$B{W}_{k}=\{\begin{array}{cc}BW/{2}^{k}& (1\le k\le n)\\ BW/{2}^{n}& (k=n+1)\end{array}$$ $${N}_{k}=\{\begin{array}{cc}N/{2}^{k}& (1\le k\le n)\\ N/{2}^{n}& (k=n+1)\end{array}$$ The bandwidth of, and number of samples in each subband (except the last) is half those of the previous subband. The last two subbands have the same bandwidth and number of samples since they originate from the same level in the filter bank. |

| All output subbands have a sample period of 2 | Sample period of kth output $$=\{\begin{array}{cc}{2}^{k}({T}_{si})& (1\le k\le n)\\ {2}^{n}({T}_{si})& (k=n+1)\end{array}$$ Due to the decimations by 2, the sample period of each subband (except the last) is twice that of the previous subband. The last two subbands have the same sample period since they originate from the same level in the filter bank. |

| The total number of samples in all of the output subbands is equal to the number of samples in the input (due to the decimations by 2 at each level). | |

| In wavelet applications, the highpass and lowpass wavelet-based filters are designed so that the aliasing introduced by the decimations are exactly canceled in reconstruction. |

Dyadic synthesis filter banks are constructed from the following basic unit. The unit can be cascaded to construct dyadic synthesis filter banks with either a asymmetric or symmetric tree structure as illustrated in the figures entitled n-Level Asymmetric Dyadic Synthesis Filter Bank and n-Level Symmetric Dyadic Synthesis Filter Bank.

Each unit consists of a lowpass (LP) and highpass (HP) FIR filter
pair, preceded by an interpolation by a factor of 2. The filters are
halfband filters with a cutoff frequency of *F _{s}* / 4, a quarter of the input sampling
frequency. Each filter passes the frequency band that the other filter
stops.

The unit takes in adjacent high-frequency and low-frequency subbands, and reconstructs them into a wide-band signal. Compared to each subband input, the output has twice the bandwidth and twice the sample rate.

The following figures illustrate the *concept* of
a filter bank, but *not* how the block implements
a filter bank; the block uses a more efficient polyphase implementation.

**n-Level Asymmetric Dyadic Synthesis Filter Bank**

Use the above figure and the following figure to compare the two tree structures of the dyadic synthesis filter bank. Note that in the asymmetric structure, the low-frequency subband input to each level is the output of the previous level, while the high-frequency subband input to each level is an input to the filter bank. In the symmetric structure, both the low- and high-frequency subband inputs to each level are outputs from the previous level.

**n-Level Symmetric Dyadic Synthesis Filter Bank**

The following table summarizes the key characteristics of symmetric and asymmetric dyadic synthesis filter banks.

**Notable Characteristics of Asymmetric and Symmetric
Dyadic Synthesis Filter Banks**

Characteristic | N-Level Symmetric | N-Level Asymmetric |
---|---|---|

| Both the high-frequency and low-frequency input subbands to each level (except the first) are the outputs of the previous level. The inputs to the first level are the inputs to the filter bank. | The low-frequency subband input to each level (except the first) is the output of the previous level. The low-frequency subband input to the first level, and the high-frequency subband input to each level, are inputs to the filter bank. |

| 2 | n+1 |

| All inputs subbands have bandwidth BW / 2 | For an output with bandwidth BW and N samples, the $$B{W}_{k}=\{\begin{array}{cc}BW/{2}^{k}& (1\le k\le n)\\ BW/{2}^{n}& (k=n+1)\end{array}$$ $${N}_{k}=\{\begin{array}{cc}N/{2}^{k}& (1\le k\le n)\\ N/{2}^{n}& (k=n+1)\end{array}$$ |

| All input subbands have a sample period of 2 T._{so} | Sample period of $$=\{\begin{array}{cc}{2}^{k}({T}_{so})& (1\le k\le n)\\ {2}^{n}({T}_{so})& \left(k=n+1\right)\end{array}$$ where the output
sample period is |

| The number of samples in the output is always equal to the total number of samples in all of the input subbands. | |

| In wavelet applications,
the highpass and lowpass wavelet-based filters are carefully selected
so that the aliasing introduced by the decimation in the dyadic |

For more information, see Dyadic Synthesis Filter Bank.

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