DTFT, DFT, ZT, and FFT 489
a. The MATLAB command [zeros, poles, k] = tf2zp(G, F) returns the zeros, poles, and
gain k of a given transfer function given by G and F, two row vectors consist-
ing of coeffi cients of the polynomials of G(z) and F(z) arranged in descending
powers of z, where H(z) = G(z)/F(z).
b. The MATLAB command [G, F] = zp2tf(zeros, poles, k) returns the coeffi cients of
the polynomials representing the numerator (G) and denominator (F) of H(z)
arranged in descending powers of z, given the zeros, poles, and gain k of H(z).
R.5.84 Recall that the plot of the poles and zeros are shown on the z-plane (complex plane),
including the unit circle drawn automatically as reference, by using the following
MATLAB commands:
a. zplane(zeros, poles), where the zeros and poles are given as column vectors
b. zplane(G, F), where G and F are row vectors consisting of the coeffi cients of the
polynomials of G(z) and F(z) arranged in descending powers of z representing
the numerator (output) and denominator (input) of H(z), respectively
R.5.85 Recall that the command [res, poles, k] = residuez(G, F) returns the partial fraction
expansion coeffi cients of H(z), where G and F are row vectors consisting of the coef-
fi cients of the polynomials of G(z) and F(z) arranged in descending powers of z,
representing the numerator and denominator of H(z), respectively.
R.5.86 Recall that the command [G, F] = residuez(res, poles, k) returns G and F, in the form
of row vectors consisting of the coeffi cients of the polynomials of G(z) and F(z)
arranged in descending powers of z, representing the numerator and denominator,
respectively, of H(z); given as column vectors res, poles, and k, referred in R.5.85.
R.5.87 Recall that the output of a discrete system can be evaluated by using the following
two MATLAB commands:
a. [h, n] = impz(G, F, leng) returns the discrete system impulse response h, given the
vectors G and F, consisting of the coeffi cients of the polynomials of G(z) and F(z)
arranged in descending powers of z (representing the numerator and denomi-
nator of H(z), respectively), and leng is an optional parameter that defi nes the
desired length of h.
Recall that the impulse response is given by Z−^1 [H(z)].
b. gn = fi lter(G, F, fn) returns the discrete system output gn, where G and F are the
coeffi cients of the polynomials G(z) and F(z) of H(z), respectively, arranged in
descending powers of z.
R.5.88 DTFT as well as ZT deals with arbitrary length sequences, including infi nite length
sequences. DFT is a computational transform when spectral representation is desir-
able, dealing with a fi nite length sequence. There is a simpler computational rela-
tion between the time samples and its corresponding frequencies in the DFT.
For a fi nite sequence of length N, the DTFT{F(ejW)} of a discrete-time sequence f(n)
returns a set of N frequencies, which may be suffi cient to approximate the spec-
trum of f(n).
R.5.89 Let f(n) be an N-length fi nite complex value sequence with nonzero values over
the range 0 ≤ n ≤ N − 1 , then the DFT, denoted by F(k), is the N-point sequence
defi ned byFk f nejNnk k
nN
()()()
2
01
∑ for 0,1,2,3, ,... N^1