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ECE 438Assignment No. 8Fall 2018

1. The digital synthesizer for voiced speech shown below operates at an 8 kHz sampling rate.

The excitation is given by

.

The vocal tract transfer function has poles and zeros at the locations shown below:

a.What is the pitch period in seconds and the pitch frequency in Hz?

b.Find the formant frequencies in Hz, and rank them according to their strength, i.e. how peaked the vocal tract response is at the corresponding frequency.

c.Sketch what the speech waveform might look like.

2.Consider the spectrogram shown below for the utterance “Every salt breeze comes from the sea.”

a.Assuming the entire utterance lasted 2 sec, very roughly estimate the pitch period.

b. Is this a wideband or narrowband spectrogram?

c. If your answer to part a, was “wideband”, sketch what a narrowband spectrogram for this same signal would look like. On the other hand, if your answer to part a, was “narrowband”, sketch what a wideband spectrogram for this same signal would look like.

d. Identify the formant frequencies at the time marked by the arrow.

e.What phoneme do you think is being uttered at this point? Support your answer by comparison with the formant frequencies in the table below:

3.Consider the signal

Assume a rectangular window

a.Compute the STDTFT as defined below

for the following cases (Be sure to express your answer in terms of the function for appropriate values of ):

i.

ii.

iii.

b.Sketch for all n. Be sure to label important dimensions.

  1. In class, we derived conditions for perfect reconstruction using an -channel modulated filter bank in which the -th channel has unit sample response , where is the unit sample response of the 0-th channel. Thus the frequency response of each channel is just a shifted version of the frequency response of the 0-th channel. Consider a system with unit sample response for the 0-th channel given by

.

  1. Sketch the unit sample response for.
  2. Derive the frequency response corresponding to the unit sample response .
  3. Sketch for.
  4. Show that satisfies the time-domain condition derived in class for perfect reconstruction with a modulated filter bank.
  5. Show that satisfies the frequency-domain condition derived in class for perfect reconstruction with a modulated filter bank.
  6. Comment on the frequency selectivity of the filter bank with base unit sample response given above, compared with a filter bank having base unit sample response .

5.In this problem, we will consider the 2-channel filter bank shown below. This structure is more general than that considered in the preceding problem.

a.The filters and are referred to as the analysis filters, since they analyze the signal into its constituent frequency components (in this case, usually the low-pass and high-pass bands, respectively). The filters and are called the synthesis filters, since they are used to synthesis the complete signal from the frequency band components of . Derive conditions on the frequency responses of these four filters that guarantee perfect reconstruction with this system, i.e. .

b.Show that two ideal band-pass filters (low-pass and high-pass) with cutoff at radians/sample will satisfy these conditions.

Note that the channels at the center of the system are only shown to indicate where quantization, coding, and transmission, or storage would take place. Here we are only interested in the filter bank; so you should assume that both channels are ideal, and have no effect on the signal.