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Why do I receive different results when comparing a digital ?

Why do I receive different results when comparing a digital filter designed using the BUTTER and BILINEAR functions in the

Signal Processing

Toolbox?Using the following specifications:  

Fs     = 172800; Fc     = 70000;

where "Fs" represents the sampling frequency and "Fc" represents the cutoff frequency, I created a digital filter in two different ways:

1. Using the BUTTER function.

[B ,A] = butter(4,Fc/(Fs/2));

2. Using the BILINEAR function to convert the analog filter designed using the BUTTER function into a digital filter through a bilinear transformation.

[Bs,As] = butter(4,Fc*2*pi,'s'); [Bz,Az] = bilinear(Bs,As,Fs);

I then used the FVTOOL function to compute the magnitude response and compared the two results:  

fvtool(B,A,Bz,Az);

Notice that the results of the two magnitude responses are not the same even though I am using the same specifications.

ANSWER

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This is the expected behavior for filter discretization in the

Signal Processing

Toolbox. The example above illustrates the nonlinear nature of the bilinear transformation. To counteract this nonlinearity, it is necessary to create analog domain filters with "prewarped" band edges, which map to the correct locations upon bilinear transformation. If you want to design the analog filter that will be equivalent to the design using the ‘s’ option in the BUTTER function, you need to prewarp the frequency as follows:

Wc = tan(pi/2*Fc/(Fs/2)); % Analog prewarped freq. rad/second

Note that this frequency is already in rad/second so the analog design is:  

[Bs,As] = butter(4,Wc,'s');

Now the issue is what sampling frequency (Fs) to use with BILINEAR. In the prewarping method used to compute "Wc", we implicitly used the bilinear transformation that is most commonly seen in the literature:

s=(1-z^(-1))/(1+z^(-1)).

In the BILINEAR function documentation, the following transformation is used:

SEE COMPLETE ANSWER CLICK THE LINK

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Digital Signal Processing, as the name suggests, is a branch of Signal Processing, which deals with signals encoded in the form of AC/DC discrete/continuous time-based pulse signals. Digital Signal Processing is one of the key rudimentary courses taught at the undergraduate level.DSP System Assignment Help

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Why are analog signals still being used in communication?

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This is a very often misunderstood concept. Here is the most important thing to understand: The communication SIGNAL is ANALOG, but the MODULATION is DIGITAL.

Why is the signal analog?

A) Our world is analog. Signals are electromagnetic waves which are inherently analog. This can’t be changed.

2. What is modulation and why do we need it?

A) Modulation is embedding a message signal onto a CARRIER signal.

B) In the early days, the ‘message’ was voice. If everyone is trying to transmit their signal, we couldn’t hear it due to interference with every other message…. like being in a crowded room with everyone talking at once and trying to listen to a conversation across the room.

C) The ‘message’ is placed onto a CARRIER signal. The carrier is a signal (sinusoid) at a given frequency (channel) and the information is MODULATED onto the carrier. This keeps everyone’s signal from interfering if each conversation is on a separate channel.

3. What is analog vs. digital modulation?

A) Analog modulation was used first because it was easy. We had an analog signal and on the other end we wanted an analog signal. The most simplistic approach was to use AMPLITUDE MODULATION (AM). The voice message was used to change the amplitude of the carrier signal.

B) Digital modulation uses BINARY format to change the PHASE or FREQUENCY of the carrier in discrete steps (1 or 0 as in the binary number representation). That is, the frequency of the carrier will be changed by +/- some small frequency offset based on the binary message. In reality the phase is modulated but frequency may be easier to visualize and is very close.

4. Why is everything going to digital modulation over analog modulation?

A) Digital modulation is more robust. If the signal is AM modulated, then any interference or noise is added directly to the signal… so it sounds like it has ‘static’ in it.

B) Then systems went to FM instead of AM. FM = Frequency Modulation. This is still an analog modulation, but interference is far less likely to create frequency errors than amplitude errors. The FM signal can still sound like there is ‘static’ but not nearly as likely. That’s why FM sounds better than AM.

C) Digital modulation only has to be accurate enough to decode the message into a 1 or 0. High or low. So, the digital modulation is extremely robust to noise and interference… but when it breaks, it is a brick wall. It sounds great until it breaks. This can be seen if you have watched an HDTV broadcast over the air. It looks awesome until the signal degrades (storm, bad antenna, etc.) and then you see large ‘pixelation’ and it flashes in and out completely as you begin to see the pixelation.

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What’s the reason that every signal have a noise?

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Noise itself is a “signal” but instead of being deterministic and predictable (like x(t)=sin(ωt)x(t)=sin⁡(ωt)) it only can be defined and characterized through statistical parameters, such as average, variance, covariance and correlation. In signal processing, the noise is usually called a stochastic process. The most “famous” noise is white noise, which is characterized by a constant power spectrum (in the frequency domain) or, equivalently, by an autocovariance and autocorrelation equal to an impulse — δ(t)δ(t) in continuous noise, δ[n]δ[n] in discrete-time noise.

As others already said, in practice everything above absolute zero has intrinsic thermal noise; semiconductors have other types of noise (shot noise, popcorn noise, 1/f noise, …) and there is also the noise associated to electromagnetic interference from antennas, electrical appliances, etc…

So, noise is really pervasive; in practice, almost all engineering endeavors include some ways of dealing with (and be resilient and robust again) noise, or else the devices wouldn’t work properly.

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