If the system has a impulse response as h(n)=Asin(n+1)ω_{0}u(n), then the system is known as Digital frequency synthesizer.
Explanation: The given impulse response is h(n)=Asin(n+1)ω_{0}u(n).
According to the above equation, the second order system with complex conjugate poles on the unit circle is a sinusoid and the system is called a digital sinusoidal oscillator or a Digital frequency synthesizer.
The filter with the system function H(z)=z k is a:
Explanation: The system with the system function given as H(z)=z k is a pure delay system . It has a constant gain for all frequencies and hence called as All pass filter.
A comb filter is a special case of notch filter in which the nulls occur periodically across the frequency band.
Explanation: A comb filter can be viewed as a notch filter in which the nulls occur periodically across the frequency band, hence the analogy to an ordinary comb that has periodically spaced teeth.
Which of the following filters have a frequency response as shown below?
Explanation: The given figure represents the frequency response characteristic of a notch filter with nulls at frequencies at ω0 and ω1.
A digital resonator is a special two pole band pass filter with the pair of complex conjugate poles located near the unit circle.
Explanation: The magnitude response of a band pass filter with two complex poles located near the unit circle is as shown below.
The filter gas a large magnitude response at the poles and hence it is called as digital resonator.
If the low pass filter described by the difference equation y(n)=0.9y(n1)+0.1x(n) is converted into a high pass filter, then what is the frequency response of the high pass filter?
Explanation: The difference equation for the high pass filter is
y(n)=0.9y(n1)+0.1x(n)
and its frequency response is given as
H(ω)= 0.1/(1+0.9e^{jω}).
If h_{lp}(n) denotes the impulse response of a low pass filter with frequency response H_{lp}(ω), then what is the frequency response of the high pass filter in terms of H_{lp}(ω)?
Explanation: The impulse response of a high pass filter is simply obtained from the impulse response of the low pass filter by changing the signs of the odd numbered samples in h_{lp}(n). Thus
h_{hp}(n)=(1)_{n} h_{lp}(n)=(e^{jπ})^{n} h_{lp}(n)
Thus the frequency response of the high pass filter is obtained as H_{lp}(ωπ).
What is the system function for a two pole band pass filter that has the centre of its pass band at ω=π/2, zero its frequency response characteristic at ω=0 and at ω=π, and its magnitude response is 1/√2 at ω=4π/9?
Explanation: Clearly, the filter must have poles at P_{1,2}=re^{±jπ/2} and zeros at z=1 and z=1. Consequently the system function is
If the phase ϴ(ω) of the system is linear, then the group delay of the system:
Explanation: We know that the group delay of the system with phase ϴ(ω) is defined as
T_{g}(ω)=(dϴ(ω))/dω
Given the phase is linear=> the group delay of the system is constant.
An ideal filter should have zero gain in their stop band.
Explanation: For an ideal filter, in the magnitude response plot at the stop band it should have a sudden fall which means an ideal filter should have a zero gain at stop band.
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