Quadrature Mirror Filter
Elec 422 Communications Signal Processing Winter 2011 Lecture 22 Dr. Steven Blostein Department of Electrical and Computer Engineering
Kingston, Ontario
Today’s Lecture • Brief review of QMFs (so far) • Conjugate Quadrature Filters (CQF) – product filters (with Matlab example) – design procedure
Last Time Motivated by subband coding…
0
1/16 1/8
1/4
1/2
We investigated the QMF:
xˆ(n)
f
Quadrature Mirror Filter (QMF) Design Objectives 1. Eliminate aliasing caused by the decimation. The decimation creates a shifted version of the signal frequency spectrum centered at f=1/4 (ω = π rad/sample) 2. Perfect reconstruction means no aliasing, no amplitude distortion, and no phase distortion. • • 3.
Give up PR for better filters: either have amplitude or phase distortion •
4.
Aliasing cancellation not difficult Perfect reconstruction comes at the cost of poor analysis filters (defeats purpose)
No phase distortion, small amplitude distortion (Johnston filters)
Have it all: PR and good, easily designed FIR filters (CQFs)
Analysis of Quadrature Mirror Filters
Quadrature Mirror Filters: Aliasing
(by inspection)
Quadrature Mirror Filters: Amplitude and Phase Distortion
Quadrature Mirror Filters: for PR ˆ Then X(z)
Quadrature Mirror Filters: for PR
so therefore:
Quadrature Mirror Filters: for PR
(or phase distortion.) for separation into subbands.
Quadrature Mirror Filters
Although anti-symmetry also produces a linear phase FIR filter, we cannot obtain a reasonable LPF since the DC response will be zero.
Quadrature Mirror Filters Eliminating Phase Distortion (cont.)
This term must be constant.
This becomes an approximation problem. This can be solved numerically via Johnston technique (previous lecture).
Conjugate Quadrature Filters (CQFs)
than
We no longer require H0(z) to have linear phase. Substituting this new condition into our above T(z) expression…
Conjugate Quadrature Filters (CQFs) Sub
into
and linear phase.
Looking at the CQF in the time domain
sign-alternating, time-reversed
time-reversed
Product Filters
C(z) is the overall transfer function of filter bank.
Time Domain Product Filters Ignoring the delay of N samples which can be added at the end,
in order to have zero phase and constant magnitude response, e.g., perfect reconstruction.
Product Filters (cont.)
Therefore, A must be equal to 1 (see previous slide),
Or, equivalently,
Product Filters (cont.) Nonzero time component V(ω)
Resulting Product Filter F(ω) with double zeros on unit circle
Relationship Between Product Filter and Analysis Filter H0(z)
Designing a CQF:
f0(n) is a low pass filter with cutoff frequency at fc=1/4 (half-band filter) Use standard window method for designing FIR filters.
CQF Design (cont.)
CQF Design (cont.) length of f0(n)=2L-1 Transform f’0(n) =
-Aa/40
1 + 10 10
-Ap/20
-Ap/20
off of unit circle. a normalizes passband magnitude response to 1.
CQF Design (cont.) Given Aa , determine Ap and then design F0(z).
(From slide 15)
where K is a constant.
Perfect Reconstruction
Product Filter Transformation: Matlab Example
Effect of Transformation f’0(n) = f0(n) + b Note the zeros are always in reciprocal pairs
b=0.2
b=0.16
Effect of Transformation f’0(n) = f0(n) + b
b=0.12
b=0.08
Effect of Transformation f’0(n) = f0(n) + b
b=0.04
As b decreases, the zeros move away from the unit circle. In the above example, the magnitude responses have not been normalized (via parameter a).
Summary of CQF Characteristics Given a product filter, many analysis filters can be derived that have the same magnitude response: for each reciprocal zero pair, either zero could be used in the analysis filter. Usually, among all the possible analysis filters, the one with the most “linear phase” response is often chosen and nearly linear phase can sometimes be obtained. In summary, CQF design uses straightforward FIR window design techniques to obtain good filters with narrow transition widths AND guarantees perfect reconstruction. However, the CQF but does not guarantee exact linear phase analysis filters.
Kingston, Ontario
Today’s Lecture • Brief review of QMFs (so far) • Conjugate Quadrature Filters (CQF) – product filters (with Matlab example) – design procedure
Last Time Motivated by subband coding…
0
1/16 1/8
1/4
1/2
We investigated the QMF:
xˆ(n)
f
Quadrature Mirror Filter (QMF) Design Objectives 1. Eliminate aliasing caused by the decimation. The decimation creates a shifted version of the signal frequency spectrum centered at f=1/4 (ω = π rad/sample) 2. Perfect reconstruction means no aliasing, no amplitude distortion, and no phase distortion. • • 3.
Give up PR for better filters: either have amplitude or phase distortion •
4.
Aliasing cancellation not difficult Perfect reconstruction comes at the cost of poor analysis filters (defeats purpose)
No phase distortion, small amplitude distortion (Johnston filters)
Have it all: PR and good, easily designed FIR filters (CQFs)
Analysis of Quadrature Mirror Filters
Quadrature Mirror Filters: Aliasing
(by inspection)
Quadrature Mirror Filters: Amplitude and Phase Distortion
Quadrature Mirror Filters: for PR ˆ Then X(z)
Quadrature Mirror Filters: for PR
so therefore:
Quadrature Mirror Filters: for PR
(or phase distortion.) for separation into subbands.
Quadrature Mirror Filters
Although anti-symmetry also produces a linear phase FIR filter, we cannot obtain a reasonable LPF since the DC response will be zero.
Quadrature Mirror Filters Eliminating Phase Distortion (cont.)
This term must be constant.
This becomes an approximation problem. This can be solved numerically via Johnston technique (previous lecture).
Conjugate Quadrature Filters (CQFs)
than
We no longer require H0(z) to have linear phase. Substituting this new condition into our above T(z) expression…
Conjugate Quadrature Filters (CQFs) Sub
into
and linear phase.
Looking at the CQF in the time domain
sign-alternating, time-reversed
time-reversed
Product Filters
C(z) is the overall transfer function of filter bank.
Time Domain Product Filters Ignoring the delay of N samples which can be added at the end,
in order to have zero phase and constant magnitude response, e.g., perfect reconstruction.
Product Filters (cont.)
Therefore, A must be equal to 1 (see previous slide),
Or, equivalently,
Product Filters (cont.) Nonzero time component V(ω)
Resulting Product Filter F(ω) with double zeros on unit circle
Relationship Between Product Filter and Analysis Filter H0(z)
Designing a CQF:
f0(n) is a low pass filter with cutoff frequency at fc=1/4 (half-band filter) Use standard window method for designing FIR filters.
CQF Design (cont.)
CQF Design (cont.) length of f0(n)=2L-1 Transform f’0(n) =
-Aa/40
1 + 10 10
-Ap/20
-Ap/20
off of unit circle. a normalizes passband magnitude response to 1.
CQF Design (cont.) Given Aa , determine Ap and then design F0(z).
(From slide 15)
where K is a constant.
Perfect Reconstruction
Product Filter Transformation: Matlab Example
Effect of Transformation f’0(n) = f0(n) + b Note the zeros are always in reciprocal pairs
b=0.2
b=0.16
Effect of Transformation f’0(n) = f0(n) + b
b=0.12
b=0.08
Effect of Transformation f’0(n) = f0(n) + b
b=0.04
As b decreases, the zeros move away from the unit circle. In the above example, the magnitude responses have not been normalized (via parameter a).
Summary of CQF Characteristics Given a product filter, many analysis filters can be derived that have the same magnitude response: for each reciprocal zero pair, either zero could be used in the analysis filter. Usually, among all the possible analysis filters, the one with the most “linear phase” response is often chosen and nearly linear phase can sometimes be obtained. In summary, CQF design uses straightforward FIR window design techniques to obtain good filters with narrow transition widths AND guarantees perfect reconstruction. However, the CQF but does not guarantee exact linear phase analysis filters.
