Continuous, discrete Fourier transformation and Series

Http://blog.csdn.net/xuexiang0704/article/details/8260890

Continuous, discrete Fourier transformation and series are still very mixed!

1. Due to the periodicity of the discrete complex index, the discrete Fourier transformation has a periodic (2 π ). This leads to a series of differences, such as discrete multiplication: the product in the time domain is equal to the convolution in the frequency domain (at this time, the convolution is a periodic convolution !)

2. Multiplication

The product in the time domain is equal to the convolution in the frequency domain. (Taking continuous signals as an example) discrete signals are slightly different.

If the input signal is a ^ t * u (t) and the frequency is 1/[1-ae ^ (-JW)];

Now we want to use (-1) ^ t in the time domain, which is equivalent to e ^ JN π;

Since e ^ JN π is a periodic signal, we can use the Fourier transformation formula of the continuous periodic signal to first obtain its Fourier series and then obtain the Fourier transformation. We can get the Fourier transformation of the signal into a pulse function (while the discrete signal is a continuous pulse, because the Fourier series coefficient of the discrete signal is cyclical, but continuous is not of this nature );

At last, continuous signals can only move the spectrum to a finite band, while discrete signals will continue to shift (however, in fact, the spectrum is 2 π, 4 π, 6 π, and so on, which is the same as 0, therefore, the low frequency is an even number of times close to π, and the high frequency is close to the odd number of times of π ).

The conclusion is that the time domain is superior to (-1) ^ N (discrete signal), and the spectrum is a shift of π units. (Continuous limited migration)

That is, by multiplying (-1) ^ N, the low-pass filter can be converted into a high-pass filter. (For example, a system with a pulse response of h (t) is a low-pass filter, and a system with a pulse response of (-1) ^ N * h (t) is a high-pass filter ).

Let's take a look at the example below.

(1) A ^ N * U (N), 0 <A <1

At this time, the spectrum is

(2) A ^ N * U (N), a <0

It is found that the spectrum is moved about π units. (The filter is changed from low-pass to high-pass ).

The multiplication feature can be used to explain why the spectrum is moved.

In addition, if there is only a low-pass filter, how can we enable it to have the function of a high-pass filter? Multiply the input signal by (-1) ^ N, and the spectrum shift moves the high frequency to the low frequency, and the low frequency to the high frequency. Through the low-pass filter, the actual rate is not only the low-frequency signal, multiply the output signal by (-1) ^ N and convert the high-frequency signal to the high-frequency position.

Http://blog.csdn.net/xuexiang0704/article/details/8260890

Continuous, discrete Fourier transformation and series are still very mixed!

1. Due to the periodicity of the discrete complex index, the discrete Fourier transformation has a periodic (2 π ). This leads to a series of differences, such as discrete multiplication: the product in the time domain is equal to the convolution in the frequency domain (at this time, the convolution is a periodic convolution !)

2. Multiplication

The product in the time domain is equal to the convolution in the frequency domain. (Taking continuous signals as an example) discrete signals are slightly different.

If the input signal is a ^ t * u (t) and the frequency is 1/[1-ae ^ (-JW)];

Now we want to use (-1) ^ t in the time domain, which is equivalent to e ^ JN π;

Since e ^ JN π is a periodic signal, we can use the Fourier transformation formula of the continuous periodic signal to first obtain its Fourier series and then obtain the Fourier transformation. We can get the Fourier transformation of the signal into a pulse function (while the discrete signal is a continuous pulse, because the Fourier series coefficient of the discrete signal is cyclical, but continuous is not of this nature );

At last, continuous signals can only move the spectrum to a finite band, while discrete signals will continue to shift (however, in fact, the spectrum is 2 π, 4 π, 6 π, and so on, which is the same as 0, therefore, the low frequency is an even number of times close to π, and the high frequency is close to the odd number of times of π ).

The conclusion is that the time domain is superior to (-1) ^ N (discrete signal), and the spectrum is a shift of π units. (Continuous limited migration)

That is, by multiplying (-1) ^ N, the low-pass filter can be converted into a high-pass filter. (For example, a system with a pulse response of h (t) is a low-pass filter, and a system with a pulse response of (-1) ^ N * h (t) is a high-pass filter ).

Let's take a look at the example below.

(1) A ^ N * U (N), 0 <A <1

At this time, the spectrum is

(2) A ^ N * U (N), a <0

It is found that the spectrum is moved about π units. (The filter is changed from low-pass to high-pass ).

The multiplication feature can be used to explain why the spectrum is moved.

In addition, if there is only a low-pass filter, how can we enable it to have the function of a high-pass filter? Multiply the input signal by (-1) ^ N, and the spectrum shift moves the high frequency to the low frequency, and the low frequency to the high frequency. Through the low-pass filter, the actual rate is not only the low-frequency signal, multiply the output signal by (-1) ^ N and convert the high-frequency signal to the high-frequency position.