The unit sample response h n : From 4. The system is anti-causal. In general, for phase plots, we do get non-zero phase values when the magnitudes are zero. Clearly these phase values have no meaning and should be ignored. This happens because of a particular algorithm used by Matlab. We avoided this problem by using the find function. The effect of periodicity doubling is in the doubling of magnitude of each sample.
We can now generalize this argument. The sequence y1 n is a point aliasing version on x n while y2 n is a zero-padded version of x n. Clearly, the DFT in part 1.
This approach can be used to verify a DFT function. Let N be an even integer. Specifically, let x1 n and x2 n be two N -point sequences. This can be obtained from the one given in the text by two simple changes. If m is a scaler then y n is circularly shifted sequence or array.
If m is a vector then y n is a matrix, each row of which is a circular shift in x n corresponding to entries in the vector m. Sequence x n in a closed form: The time-domain sequence x n is a linear combination of the harmonically related complex exponential. Using the results from Example 5. Such a matrix is called a circulant matrix. It is completely described by the first column or the row. Circular Convolution x3 n 1 Amplitude 0. Circular Convolution x3 n Amplitude 4 2 0 0 5 10 15 20 25 30 35 40 45 50 n Figure 5.
Circular Convolution: x3 n 1. In this method we have to save the intermediate convolution results and then properly overlap these before adding to form the final result y n. From the results of the above two parts, the minimum value of N to make the circular convolution equal to the linear convolution is 7. If we make N equal to the length of the linear convolution which is equal to the length of x1 n plus the length of x2 n minus one, then the desired result can be achieved.
This is a windowed cosine sequence containing no leakage. Verification of the results of parts 1. Sequence: x4 n 1 0. Sequence: x5 n 1 0. It is a natural result due to the fact that bandlimited periodic cosines are sampled over noninteger periods. Due to this fact, the periodic extension of x n does not result in a continuation of the cosine waveform but has a jump at every N interval.
This jump results in the leakage of one frequency into the abducent frequencies and hence the result of the Problem P5. Sequence: x n 1 0. This is a windowed sine sequence containing no leakage. Sequence: x2 n 1 0. The above result is the leakage property of sines.
It is a natural result due to the fact that bandlimited periodic sines are sampled over noninteger periods. Due to this fact, the periodic extension of x n does not result in a continuation of the sine waveform but has a jump at every N interval.
An N -point DFT is used to obtain an estimate of the magnitude spectrum of xa t. Consider the overlap-and-save method of block convolution along with the FFT algorithm to implement high-speed block convolution. Using this approach, determine y n with FFT sizes of , , and Compare the above approaches in terms of the convolution results and their execution times.
Canonical structure: i The given structure is canonical ii The given structure is not canonical. The canonical structure is 0. Block diagram of the above system with input node x n and output node y n is shown below.
Transposed block diagram: The block diagram due to steps i and ii is shown below. The normal direct form I structure block diagram is shown below on the left. The transposed direct form I structure block diagram is shown above on the right.
The normal direct form II structure block diagram is shown below on the left. The transposed direct form II structure block diagram is shown above on the right. Clearly it looks similar to that in part 1. A parallel structure containing second-order normal direct from II sections: Matlab script: 1.
A parallel structure containing second-order transposed direct from II sections: Matlab script: 1. The required form is the cascade form. This solution is not unique since numerator and denominator biquads can be grouped differently. The given signal flow graph is a parallel connection containing one second-order cascade branch. The solution is not unique since any two out of three parallel biquads can be used to construct a cascade branch.
Block diagram: 3. Due to a mistake in labeling, two of the multiplier coefficients in this structure have incorrect values rounded to 4 decimals. To locate these two multipliers and determine their correct values we will investigate their pole-zero structure and combine the appropriate pairs.
An overall cascade structure containing second-order section and which contains the least number of multipliers: This can be obtained by combining pole or zero pairs with the same magnitude but differing signs. Hence the phase response is still linear and is given by 6. The sum in 6.
The frequency sampling structure for the impulse response given in Example 6. Example 6. The block diagram of the lattice form is y n 2 0. Cascade-2 5 5 iv.
Cascade-3 5 5 v. This implies that the filter is a linear-phase FIR filter. Clearly the first four plots satisfy the zero-placement requirements and hence the corresponding filters are linear-phase filters. Plot amplitude the response of resulting filter. Compare your results. Ideal Impulse Response Hann Window 0. Ideal Impulse Response Hamming Window 0. The filter response plots are shown in Figure 7.
We want to approximate this filter using a frequency sampling design in which we choose 40 samples. Determine the minimum stopband attenuation. The bandwidth should be no more than 0.
Use the optimum value for transition band samples and draw the frequency sampling structure. Determine the impulse response h n and draw the linear-phase structure. FIR2 Function Design 2. Amplitude Response 1 0. Impulse Response: Bandpass 0. From Figure 7. The order using Parks-McClellan algorithm is Impulse Response 0. Amplitude Response 2 Amplitude 1 0 0. Amplitude Response 3. Amplitude Response in Part 1 4. Amplitude Response in Part 3 4. Amplitude Response 4.
Thus this is an optimum design. Talha Arshad. Get the pass band and stop band ripples. Get the pass band and stop band edge frequencies.
Get the sampling frequency. Find the filter coefficients. Draw the magnitude and phase responses. Formula The transfer function of the Butterworth reference analog prototype filter is expressed as follows: where: Sk is the k-th pole of the Butterworth filter transfer function, N is the filter order.
Get the pass band and stop band ripple. Calculate hd n , hw n , h n for various values for rectangular window. Calculate hd n , w n , h n for various values for hamming window. Calculate hd n , hw n , h n for various values of Kaiser window.
Define the value of n. Find the fast fourier transform. Find the complex conjugate of fft. Theory The spectrum of a signal is fundamentally a description of the energy of a signal as a function of frequency. It may also be a function of time and frequency which leads to the time-varying or the so-called time-frequency spectrum description.
Spectra are represented by their amplitude and phase or more meaningfully, by the squared magnitude, which is referred to as the Power Spectrum. Define the values of n,n1,etc. Split the given function for step of 1 or 0. Calculate a signal that is periodic with respect to the input signal. Plot the graph of the given input signal and periodic signal. Calculate a signal that is a periodic with r to the input signal. Plot the graph of given input signal and aperiodic signal.
Get the input for the signal 2. Calculate the required parameter for first order frequency response. Plot the given input signal. Plot the graph of magnitude spectrum. Plot the graph of phase spectrum.
Get the input for n. Calculate x and assign it to ones matrix for length of N. Find the analytic response of the system for the given equation. Plot in graph the input and step response of the positive output. Compute the frequency response of Z transform of Unit circle 3.
Plot the Frequency Response. The function impz provides the samples of the time domain sequence, 2. Where num, den represent vectors containing numerator and denominator coefficients of z-transform. The length of output y is equal to input x. If an impulse input sequence is passed to the z-transform , the output will be the inverse z-transform 6. Such parallelism supports a powerful set of arithmetic, logic, and bit-manipulation operations that can all be performed in a single machine cycle.
It also includes the control mechanisms to manage interrupts ,repeated operations, and function calling. The PLU performs Boolean operations or the bit manipulations required of high-speed controllers. This makes instruction to executes faster compared to micro processors. The clock generator can be driven internally by a crystal resonator. The timer can be stopped, restarted, reset, or disabled by specific status bits.
This feature consists of multiple wait state generating circuits. Each circuit is user-programmable to operate in different wait states for off-chip memory accesses. The BSP provides flexibility on the data stream length. Data is framed either as bytes or as words. The ADSP Family processors are single-chip microcomputers optimized for digital signal processing DSP and other high speed numeric processing applications. The ADSPxx processors are all built upon a common core.
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