Discrete-Time Fourier Transform Discrete Fourier Transform z-Transform

239 Pages·2008·1.12 MB·English

Discrete-Time Fourier Transform
Discrete Fourier Transform
z-Transform
Tania Stathaki
811b
Joseph Fourier (1768-1830)
Discrete-Time Fourier Transform
• Definition - The Discrete-Time Fourier
jω
Transform (DTFT) X ( e ) of a sequence
x[n] is given by
∞
jω − jω n
X (e ) = x[n]e
∑
n=−∞
jω
• In general, X ( e ) is a complex function
of the real variable ω and can be written as
jω jω jω
X (e ) = X (e ) + j X (e )
re im
Discrete-Time Fourier
Transform
jω jω
• X (e ) and X ( e ) are, respectively,
re im
jω
the real and imaginary parts of X ( e ) , and
are real functions of ω
jω
• X (e ) can alternately be expressed as
jω jω jθ(ω)
X (e ) = X (e )e
where
jω
θ(ω) = arg{X (e )}
Discrete-Time Fourier
Transform
jω
• X (e ) is called the magnitude function
• θ(ω) is called the phase function
• Both quantities are again real functions of ω
• In many applications, the DTFT is called
the Fourier spectrum
jω
• Likewise, X ( e ) and θ ( ω ) are called the
magnitude and phase spectra
Discrete-Time Fourier
Transform
2
jω jω ∗ jω
X (e ) = X (e )X (e )
jω jω
X (e ) = X (e ) cosθ (ω)
re
jω jω
X (e ) = X (e ) sinθ (ω)
im
2
jω 2 jω 2 jω
X (e ) = X (e ) + X (e )
re im
jω
X (e )
im
tanθ (ω) =
jω
X (e )
re
Discrete-Time Fourier Transform
jω jω
• For a real sequence x[n], X (e ) and X ( e ) are
re
jω
even functions of ω, whereas, θ ( ω ) and X ( e )
im
are odd functions of ω (Prove using previous slide relationships)
jω jω jθ(ω+2πk)
• Note: X (e ) = X (e )e
jω jθ(ω)
= X (e )e
for any integer k
• The phase function θ(ω) cannot be uniquely
specified for any DTFT
Discrete-Time Fourier
Transform
• Unless otherwise stated, we shall assume
that the phase function θ(ω) is restricted to
the following range of values:
− π ≤ θ(ω) < π
called the principal value
Discrete-Time Fourier
Transform
• The DTFTs of some sequences exhibit
discontinuities of 2π in their phase
responses
• An alternate type of phase function that is a
continuous function of ω is often used
• It is derived from the original phase
function by removing the discontinuities of
2π
Discrete-Time Fourier
Transform
• The process of removing the discontinuities
is called “unwrapping”
• The continuous phase function generated by
unwrapping is denoted as θ (ω)
c
• In some cases, discontinuities of π may be
present after unwrapping