Q1. Fourier transform
Figure Q1:
(a) The plot Q1 shows the frequency spectrum of a short sentence spoken into a micro-
phone with flat frequency response.
i. Do you expect just one fundamental frequency and its harmonics – or multiple
fundamental frequencies and their harmonics? Provide an explanation and state
at least two possible fundamental frequencies. [6]
ii. Speech has a-periodic hissing sounds such as “s”, “z” or “f”. Provide an ap-
proximate frequency range in the plot and an explanation why these sounds
generate no distinct peaks in plot Q1. [6]
(b) The sound recording of the speech is 2 secs long at a sampling rate of 50 KHz.
Your task is to improve the quality of the speech by boosting the amplitudes of the
high order harmonics in the frequency domain by manipulating the Fourier trans-
formed frequency coefficients X [k] and then transforming them back into the time
domain. Calculate the index range of k1 . . .k2 of the frequency samples based on the
frequency range of the harmonics F1, . . .F2 and provide the formula(s) or code to
boost the corresponding frequency amplitudes by a factor of two so that the inverse
becomes real valued again. [8]
(c) After having performed the inverse Fourier Transform you try to save the time series
as a WAV file but the function crashes because the time series is complex. Provide
one reason why it is the case and what can be done to fix it. [5]
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Q2. FIR filtering
(a) An electrocardiogram (ECG) has an unwanted DC drift. Which filter should be
applied and what is the cut-off frequency considering the heart rate of a healthy
person. [5]
(b) An ECG is sampled at a rate of 250 Hz and filtering needs to be done with an FIR
filter. State the theoretical number of taps and the actual number required for DC
removal as implemented in the lab. [10]
(c) State at least one strategy how FIR filters can be made more computationally effi-
cient. [5]
(d) The discrete convolution is defined as:
y(n) = h(n)∗ x(n) =
∞
∑
n=−∞
h(n)x(m−n)
A Google search often claims that an FIR filter performs this operation. Is that
statement 100% true if we don’t know anything about h(n) or x(n)? Provide an
explanation. [5]
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Q3. A digital filter has the transfer function:
H(z) =
(1− r0e
jω0z−1)(1− r0e
− jω0z−1)
(1− r1e jω1z−1)(1− r1e− jω1z−1)
(a) Draw a dataflow diagram of the filter. [8]
(b) Which variable in the formula above determines the stability of the filter and which
values guarantee a stable operation? [2]
Figure Q3:
(c) Butterworth filter coefficients of even filter order are evenly distributed in the left
half s plane forming a half circle as shown in Fig. Q3. For example a 6th order filter
has 3 pairs of poles as shown in Fig. Q3 where the imaginary values flip sign. How
would you distribute the poles over a chain of the above filters so that processing is
guaranteed real valued within every building block and that higher order filters can
be created with minimal additional algebra? [10]
(d) Assume the analogue filter is a lowpass filter, how would you transform the analogue
poles of Fig. Q3 to the digital coefficients of the above transfer function? State the
transform and provide a reason why it is appropriate. [5]
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Q4. FIR filter design in the frequency domain
(a) Sketch the frequency response of an ideal digital bandstop filter with cut off fre-
quencies ω1 and ω2. [5]
(b) Perform an inverse Fourier transform of the ideal bandstop filter to obtain the im-
pulse response of the filter. [10]
(c) Your task is to design a narrow bandstop filter. Which window function would you
choose that maximises the bandstop rejection? [5]
(d) You sample a signal at 250Hz and the delay of the bandstop filter needs to be less
than 80ms. What is the maximum number of taps permitted? [5]
Page 5 of 5 /END