Circuits Syst Signal Process (2013) 32:3131–3134
DOI 10.1007/s00034-013-9604-5
S H O RT PA P E R
A Note on Generalized Linear Phase Conditions
for Causal FIR Systems
S.C. Dutta Roy
Received: 18 September 2012 / Revised: 18 April 2013 / Published online: 3 May 2013
© Springer Science+Business Media New York 2013
Abstract An alternative proof is presented for the necessary conditions for generalized linear phase in causal FIR systems, which appears to be somewhat crisper and
more straightforward than the existing proofs available in the literature.
Keywords Generalized linear phase conditions · FIR systems · Digital signal
processing
1 Introduction
As is well known, a causal FIR system with impulse response h(n), 0 ≤ n ≤ N − 1,
is known to have generalized linear phase when h(n) = ±h(N − 1 − n). In general,
the frequency response H (ej ω ), which is the Fourier transform (FT) of h(n), is
−1
N
H ej ω =
h(n)e−j ωn
(1)
n=0
If H (ej ω ) has generalized linear phase, then it has the general form
H ej ω = A(ω)e−j ωα+jβ
(2)
where A(ω), the so-called “pseudo magnitude”, is real and could be positive or negative ; α = (N − 1)/2 ; and β = 0 or π for symmetric and ±π/2 for asymmetric h(n).
Most text books on digital signal processing demonstrate the sufficiency of these conditions. That these conditions are also necessary has been dealt with by only a few
S.C. Dutta Roy ()
Dept. of Electrical Engg., IIT Delhi, Hauz Khas, New Delhi 110 016, India
e-mail: s.c.dutta.roy@gmail.com
Present address:
S.C. Dutta Roy
164, SFS(DDA) Apartments, Hauz Khas, New Delhi 110 016, India
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Circuits Syst Signal Process (2013) 32:3131–3134
text book authors [1–7], and also in a recent paper [8]. In the next section, we give
a brief review of these methods. In the following section, an alternative proof of necessity is presented, which appears to be somewhat crisper and more straightforward
than the known ones.
2 Review of Previous Work
In [7] and [5], the authors use Fourier series properties for the aforementioned proof,
but do not explain how the possible values of β are arrived at. In [1] and [4], the possible values of β are first derived and then the symmetry and asymmetry conditions
are obtained from the fact that A(ω) is real and that A(ω) = ±A(−ω). In [6], the author does not assume that h(n) is FIR to start with, and invokes periodicity of H (ej ω )
to show that 2α must be an integer. He then follows essentially the same arguments
as in [1] and [4] to find the possible values of β. In the next step, he finds h(2α − n)
by inverse FT and shows that it is equal to ±h(n). Finally, causality is introduced to
show that h(n) must be finite with α = (N − 1)/2 and h(n) = ±h(N − 1 − n). The
distinctive feature of [2] is that it does not assume h(n) to be real to start with. The
authors use inverse FT to establish a relation between h(n) and a(n), the inverse FT
of A(ω), and then use the Hermitian symmetry property of a(n) to derive a general
relation between h(n) and h∗ (N − 1 − n) where ∗ denotes complex conjugate. In the
process, they take α = (N − 1)/2 without adequate justification. Finally, they invoke
realness of h(n) to find the possible values of β and the symmetry and antisymmetry
conditions. In [3], which is a later edition of [2], the authors take α = (N − 1)/2 by
a somewhat “weaker argument” as commented in [8]. In the method of [8], h(n) is
assumed real, and it is first proved that the group delay must be an integer or half
integer, by invoking the periodicity of H (ej ω ), as in [6]. Next, essentially the same
arguments are used as in [1] and [4] to find the possible values of ejβ . However, instead of using the specific values of β, as in [6], the sample h(2α − n) is considered
and it is demonstrated that it must be equal to ej 2β h(n), which translates to symmetry
or antisymmetry of h(n).
3 Alternative Proof
The alternative proof we present here looks at the problem from a different and a
direct approach as compared to the other methods reviewed in the previous section.
Following the earlier works [1] and [4], we first establish the possible values of β.
Specifically, real h(n) implies that H (ej ω ) must be Hermitian symmetric, i.e.,
(3)
H ej ω = H ∗ e−j ω
where ∗ denotes complex conjugate. Combining (3) with (2), we get
A(ω)e−j ωα+jβ = A(−ω)e[j (−ω)α−jβ]
(4)
which implies that A(ω)ej 2β = A(−ω). Since A(ω) is real, ej 2β must also be real,
so
ej 2β = ±1
(5)
Circuits Syst Signal Process (2013) 32:3131–3134
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Therefore, for β = 0 or π , A(ω) = A(−ω) (even) and for β = π/2 or 3π/2, A(ω) =
−A(−ω) (odd). The next step is to combine (1) and (2) to get
A(ω) = e−jβ
N
−1
h(n)e−j ω(n−α)
(6)
n=0
Let the summation in (6) be denoted by G(ej ω ). Also let m = n − α and g(m) =
h(m + α). Then
−1−α
j ω N
G e
=
g(m)e−j mω
(7)
m=−α
When α is an integer, G(ej ω ) is easily identified as the FT of the sequence g(m),
which is the same as h(n) shifted to the left by α samples. When α is not an integer,
G(ej ω ) is like the FT of g(m) with the difference that m takes non-integer values, but
at intervals of unity. If β = 0 or π , then ejβ is ±1 and since A(ω) is real, G(ej ω ) must
also be real, which is possible only if g(m) is an even sequence starting at m = −α
and ending at m = N − 1 − α; so
α = (N − 1)/2
(8)
Also, g(m) = g(−m) or g(n − α) = g(α − n) which implies that
h(n) = h(2α − n) = h(N − 1 − n)
(9)
i.e., the impulse response is symmetric. On the other hand, when β = ±π/2,
=
±j and in order to have real A(ω), (6) dictates that G(ej ω ) must be purely imaginary,
which is possible only if g(m) is an odd sequence starting at m = −α and ending at
m = N − 1 − α. As in the previous case, this implies (8) and g(m) = −g(−m) gives
ejβ
h(n) = −h(N − 1 − n)
(10)
i.e., h(n) is antisymmetric. This completes the proof.
4 Conclusions
An alternative proof has been presented in this paper of the generalized linear phase
condition by using the facts that the FT or FT-like summation of an even sequence is
purely real and that of an odd sequence is purely imaginary. This proof appears to be
crisper and simpler than the proofs available in the existing literature.
Acknowledgements The author thanks Professor P.S.R. Diniz for supplying the relevant pages from his
book [3] as well as its earlier edition [2] and Dr. Sumantra Dutta Roy for his help in preparing this paper.
The author also thanks the reviewers for their careful reading of the manuscript and offering comments
and suggestions which helped in improving the presentation.
References
1. T.J. Cavicchi, Digital Signal Processing (Wiley, New York, 2000)
2. P.S.R. Diniz, E.A.B.D. Silva, S.I. Netto, Digital Signal Processing: System Analysis and Design (Cambridge University Press, Cambridge, 2002)
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3. P.S.R. Diniz, E.A.B.D. Silva, S.I. Netto, Digital Signal Processing: System Analysis and Design (Cambridge University Press, Cambridge, 2010)
4. S.K. Mitra, Digital Signal Processing—A Computer Based Approach (McGraw-Hill, New York, 2011)
5. A.V. Oppenheim, R.W. Schafer, Discrete Time Signal Processing (Pearson, Boston, 2010)
6. B. Porat, A Course in Digital Signal Processing (Wiley, New York, 1997)
7. L.R. Rabiner, B. Gold, Theory and Applications of Digital Signal Processing (Prentice Hall, Englewood Cliffs, 1975)
8. S. Zhang, Note on generalized linear phase property of discrete time finite impulse response systems.
Circuits Syst. Signal Process. 31(1), 385–388 (2012)