Like continuous time signal Fourier transform, discrete time Fourier Transform can be used to represent a discrete sequence into its equivalent frequency domain representation and LTI discrete time system and develop various computational algorithms. T, is a continuous function of x n. Hence, this mathematical tool carries much importance computationally in convenient representation. Both, periodic and non-periodic sequences can be processed through this tool.
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Objective: In this project, you will perform Fourier analysis of various temporal waveforms using RF. In this tutorial you will learn how to use RF. You will also learn how to define arbitrary temporal waveforms.
A discrete-time Fourier transform DTFT is applied to a discrete-time signal, which can be constructed from the samples of a continuous signal.
Let's define the Fourier transform of a continuous signal x t of the time variable t in the following form:. If you sample the continuous signal x t with a sampling period of T, its samples can be represented by a discrete-time sequence in the integer variable n:. Another interpretation of the DTFT is the Fourier transform of the impulse train made up of the samples of x t :. These blocks all have a sampling period parameter T in seconds.
The DFT blocks also have another parameter N which is the sequence length or the number of discrete-time samples. The Fourier transform of a real-valued signal is typically a complex-valued function of the frequency. The DTFT can be written as:. All of RF. It is very important for you to understand that the two output signals of the DTFT blocks are still constructed and plotted as functions of time. As a result, you can always see and plot only the positive frequency axis.
Place and connect the parts as shown in the figure below. The Complex Modulus Block is used to output the absolute value or magnitude of the complex-valued spectral signal.
To generate a single pulse waveform of duration 8s, use the following parameters for the voltage source V The simulation results are shown in the figure below.
As you would have expected, the output is a periodic sinc function. The figure below shows a zoomed-in version of the above figure over the time interval [10s, 12s] so you can better see the details of the waveforms. Here you will interpret the horizontal axis as the frequency f-axis for the signals v 2 , v 3 and v 4. Since the DTFT block samples the input signal every one second, you could have also input a pulse train of period 1s with 8 pulses instead. The outputs would have been identical.
Note that in each period, the sinc function has 8 peaks. Note that the 8-point DTFT block X1 will collect a total of 8 samples of your input signal v 1 at half-second intervals. This means that the DTFT block will see your input pulse waveform only over the time interval [0, 4s].
In other words, your input signal has been effectively reduced to:. The figure below shows the new simulation results. This is expected from the scaling property of the Fourier transform. Note that the sinc function still has 8 peaks in each period.
Using RF. Spice's arbitrary temporal waveform generator, you can define any mathematical function of time as the waveform of your voltage source. In the property dialog of this source, click the Edit Model In other words, v t stands for the time variable. For this purpose, you need to enter the following expression in the arbitrary source's property dialog:. Place and connect the part as shown in the figure below.
This will provide an 8s window to the sinc function. As you would have expected, the output is a periodic pulse function, or the periodic replicas of the "rect" function with a unit width and a period of 2Hz:. It is important to note that your finite-duration input signal as seen by the DTFT block's 8s time-limited window is indeed given by the following more sophisticated function:. The above equation explains the fluctuations and aliasing effects you can clearly see in the above plots.
Your input signal is now:. Run another transient test of your new circuit with the same parameters as before. The results are shown in the figure below, which hasn't change much compare to the previous graph.
The convolving frequency-domain sinc has been compressed more tightly. As a result, the reconstructed frequency-domain pulse output is cleaner and sharper. This reduces the width of the DTFT block's sampling window back to 8s.
Run one more transient test of your circuit with the same parameters as before. In this part of the tutorial lesson, you are going to convolve a pulse function with itself. For the input pulse you will use a Finite Sequence Pulse Generator which can be accessed from the same menu as the pulse train generator.
To generate a pulse sequence of duration 8s, use the following parameters for the voltage source V From signal theory, we know that the output of the convolution is a triangular signal with a duration of 16s. Before running the simulation, open RF. Sometimes, the simulation may not converge quickly using the "trap" method. The simulation results are shown in the figure below, where you can see the resulting triangular function of double width.
Set the value of the sampling period T of this block equal to 0. The total input signal duration is still 8s, but you have twice as many input samples this time.
In this part of this tutorial lesson, you are going to take the discrete-time output signal of the convolution circuit of the previous part and pass it through DFT16 and DTFT16 blocks to calculate both the discrete-time Fourier transform DTFT and discrete Fourier transform DFT of the triangular pulse sequence. You can see that the Fourier transform results stabilize after a quite long period of time.
Zoom in the above graph and scale the time axis to the interval [ As you can see from the figures, your input signal is a triangular pulse sequence which is the result of convolution of a pulse sequence with itself. According to Parseval's theorem:. It can take both positive or negative values. Before closing this tutorial lesson, let's see what happens if you increase the DFT's sequence length or period N to Remember that on the time axis this mean signal details on a time scale of This is the value that you will choose for the sampling period of the signal hold block.
Zoom in the graph's time axis and scale it to the interval [16s, 17s] to see the reconstructed spectral response. Back to RF. Jump to: navigation , search. Navigation menu Personal tools Log in. Namespaces Page Discussion. This page was last modified on 8 November , at This page has been accessed 3, times. The Fourier transform of the continuous-time signal x t. The discrete-time Fourier transform of the discrete-time signal x[n]. The Fourier series of the periodic version of the continuous-time signal x t.
The discrete Fourier transform DFT of the discrete-time signal x[n]. For practical computation of DTFT transforms, you should keep the sampling period on the order of seconds. The zoomed-in graph of the input and output signals over the time interval [10s, 12s].
The property dialog of the Arbitrary Temporal Waveform Generator. The property dialog of the Finite Sequence Pulse Generator. A simple circuit to test the Point Discrete Convolution Block. The DFT16 block followed by a discrete-time signal hold block. The zoomed-in graph of the input and output signals of the discrete-time signal hold block in the time interval [16s, 17s].
Digital Signal Processing - DFT Introduction