**J-DSP Lab 3:
Frequency Responses and Pole-Zero Plots**

Lab 3 concentrates on generating
frequency responses and pole-zero plots from transfer functions of systems.
J-DSP contains a ** PZ Placement** block under the

**Problem 3-1: Pole-Zero Plots**

Find the poles and zeros of the
following transfer functions and use the J-DSP editor to plot the magnitude and
phase of the frequency response. Observe
the structure of poles and zeros in each system relative to the frequency
response.

a)

_{}

Is the system stable?

b)

_{}

Determine the zeros and plot the frequency response.

c)

_{}

Note the pole locations.
What kind of filter is this?

**Problem 3-2: Poles and Zeros to
Frequency Responses **

Consider a system that has the
complex conjugate poles

_{}

and a zero located at

_{}

Here

i)
r = 0.96

ii)
r = 0.71

iii)
r = 0.14

For each condition, i)-iii),
please do the following:

(a) Derive analytically the impulse response of the
system and show its dependence on r. Plot the impulse response for r=0.96 (use
J-DSP).

(b) Plot the frequency and phase response for each case
using J-DSP.

(c) Note the
differences in the frequency responses relative to the position of the poles.

**Problem 3-3: Low-pass/High-pass
Filter **

For this problem, plot the
magnitude in dB.

(a) Use the ** Filter** and the

(b) Use the ** Filter** and the

_{}

Hint: Poles raise the
frequency response up (create peaks) and zeros create valleys. Poles and zeros are entered in conjugate
pairs to get real-valued filter coefficients.

Remember that when entering zeros and poles graphically, J-DSP will compute the transfer
function automatically.

**Problem 3-4: An Interesting
Frequency Response **

Consider the following all-pass
system:

_{}

a) Use J-DSP
and find the poles and zeros of the transfer function.

b) Plot magnitude and phase responses of the system.

c) Note the symmetry of the numerator relative to the
denominator

All-pass
filters are often used to obtain design delay and phase characteristics in a
signal without altering its magnitude spectrum.