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Thus, the impulse response describes how a system behaves directly in time, while the frequency responsedescribes how it behaves in frequency. The two descriptions are intimately related because the frequency response is theFourier transform of the impulse response. This will be used repeatedly in [link] to design filters for the manipulation (augmentation or removal)of specified frequencies.

In Exercise  [link] , a system was defined to have an impulse response that is a sinc function.The Fourier transform of a sinc function in time is a rect function in frequency [link] . Hence, the frequency response of the system is a rectangle that passes all frequencies below f c = 1 / T and removes all frequencies above (i.e., the system is a lowpass filter).

M atlab can help to visualize the relationship between the impulse response and the frequency response.For instance, the system in convolex.m is defined via its impulse response, which is a decaying exponential. [link] shows its output when the input is a simple sum of delta functions, andExercise  [link] explores the output when the input is a white noise. In freqresp.m , the behavior of this system is explained by looking at its frequency response.

Ts=1/100; time=10;             % sampling interval and total time t=0:Ts:time;                   % create time vectorh=exp(-t);                     % define impulse response plotspec(h,Ts)                 % find and plot frequency response
freqresp.m numerical example of impulse and frequency response (download file)

The output of freqresp.m is shown in [link] . The frequency response of the system (which is just the magnitude spectrum ofthe impulse response) is found using plotspec.m . In this case, the frequency response amplifies low frequenciesand attenuates other frequencies more as the frequency increases. This explains, for instance, why the output of theconvolution in Exercise  [link] contained (primarily) lower frequencies, as evidenced by the slower undulations in time.

The action of a system in time is defined by its impulse response (in the top plot). The action of the system in frequency is defined by its frequency response (in the bottom plot), a kind of lowpass filter.
The action of a system in time is defined by its impulse response (in the top plot). The action of the system in frequencyis defined by its frequency response (in the bottom plot), a kind of lowpass filter.

Suppose a system has an impulse response that is a sinc function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Hint: center the sinc in time; for instance, use h=sinc(10*(t-time/2)) .

Suppose a system has an impulse response that is a sin function. Using freqresp.m , find the frequency response of the system. What kind of filter does this represent?Can you predict the relationship between the frequency of the sine wave and the location of the peaks in the spectrum?Hint: try h=sin(25*t) .

Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a sinc function (as in Exercise  [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise  [link] .

Create a simulation (analogous to convolex.m ) that inputs white noise into a system with impulse responsethat is a sin function (as in Exercise  [link] ). Calculate the spectra of the input and outputusing plotspec.m . Verify that the system behaves as suggested by thefrequency response in Exercise  [link] .

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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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