0.4 Lab 4 - sampling and reconstruction  (Page 6/7)

Upsampling a signal by a factor of $L$ is simply the process of inserting $L-1$ zeros in between each sample. The frequency domain relationship between a signal $x\left(n\right)$ and its upsampled version $z\left(n\right)$ can be shown to be the following

Therefore the DTFT of $z\left(n\right)$ is simply $X\left(e^{j\omega }\right)$ compressed in frequency by a factor of L. Since $X\left(e^{j\omega }\right)$ has a period of $2\pi$ , $Z\left(e^{j\omega }\right)$ will have a period of $2\pi /L$ . All of the original information of $x\left(n\right)$ will be contained in the interval $[-\pi /L,\pi /L]$ of $Z\left(e^{j\omega }\right)$ , and the new aliases that are created in the interval $[-\pi ,\pi ]$ are the unwanted components that need to be filtered out. In the time domain,this filtering has the effect of changing the inserted zeros into artificialsamples of $x\left(n\right)$ , commonly known as interpolated samples.

[link] shows a Simulink model that demonstrates discrete-time interpolation.The interpolating system contains three main components: an upsampler which inserts $L-1$ zeros between each input sample, a discrete-time low pass filter which removes aliasedsignal components in the interpolated signal, and a gain block to correct the magnitude of the final signal.Notice that "signal a" is the input discrete-time signal while "signal c" is the final interpolateddiscrete-time signal.

Open the experiment by double clicking on the icon labeled Discrete Time Interpolator . The components of the system are initially setto interpolate by a factor of 1. This means that the input and output signals willbe the same except for a delay. Run this model with the initial settings,and observe the signals on the Scope .

Simulink represents any discrete-time signal by holding each sample value over a certain time period.This representation is equivalent to a sample-and-hold reconstruction of the underlying discrete-time signal.Therefore, a continuous-time Spectrum Analyzer may be used to view the frequency content of the output "signal c". The Zero-Order Hold at the Gain output is required as a buffer for the Spectrum Analyzer in order to set its internal sampling period.

The lowest frequency component in the spectrum corresponds to the frequency content of the original input signal,while the higher frequencies are aliased components resulting from the sample-and-hold reconstruction.Notice that the aliased components of "signal c" appear at multiples of the sampling frequency of 1 Hz.Print the output of the Spectrum Analyzer .

Next modify the system to upsample by a factor of 4 by setting this parameter in the Upsampler . You will also need to set the Sample time of the DT filter to 0.25. This effectively increases the sampling frequency of the system to4 Hz. Run the simulation again and observe the behavior of the system.Notice that zeros have been inserted between samples of the input signal. After you get an accurate plot of the output frequency spectrum,print the output of the Spectrum Analyzer .

Notice the new aliased components generated by the upsampler. Some of these spectral components lie between the frequencyof the original signal and the new sampling frequency, 4 Hz. These aliases are due to the zeros that areinserted by the upsampler.