# Transfer Function of Two Filters in Parallel

Started by April 26, 2007
```I have two filters in parallel. The transfer functions are given by H1
and H2. I've illustrated the setup below.

.-- H1 ---.
|         |
in --------|        (+)------> out
|         |
'-- H2 ---'

How do I find the transfer function of two parallel filters?  Do I
merely sum the two together?

sum = H1+H2

I would think this would work, but I would be worried that the phase
responses would throw the result off.  How do I find the combined
transfer function?
```
```Chris Barrett <"chrisbarret"@0123456789abcdefghijk113322.none> writes:

> I have two filters in parallel. The transfer functions are given by H1
> and H2. I've illustrated the setup below.
>
>            .-- H1 ---.
>            |         |
> in --------|        (+)------> out
>            |         |
>            '-- H2 ---'
>
> How do I find the transfer function of two parallel filters?

One step at a time, using the definition of transfer function:

H(w) = Y(w) / X(w).

Start by computing the output (in frequency) of H1, Y1(w), in terms of
the input (call it X(w)). Then similarly for Y2(w).
--
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
%%% 919-577-9882                %  Verdi's always creepin' from her room."
%%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO
```
```Randy Yates wrote:
> Chris Barrett <"chrisbarret"@0123456789abcdefghijk113322.none> writes:
>
>
>>I have two filters in parallel. The transfer functions are given by H1
>>and H2. I've illustrated the setup below.
>>
>>           .-- H1 ---.
>>           |         |
>>in --------|        (+)------> out
>>           |         |
>>           '-- H2 ---'
>>
>>How do I find the transfer function of two parallel filters?
>
>
> One step at a time, using the definition of transfer function:
>
>   H(w) = Y(w) / X(w).
>
> Start by computing the output (in frequency) of H1, Y1(w), in terms of
> the input (call it X(w)). Then similarly for Y2(w).

Can I sum H1(w) and H2(w) to get H_the_system(w)?
```
```On Apr 27, 7:54 am, Chris Barrett
<"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
> I have two filters in parallel. The transfer functions are given by H1
> and H2. I've illustrated the setup below.
>
>             .-- H1 ---.
>             |         |
> in --------|        (+)------> out
>             |         |
>             '-- H2 ---'
>
> How do I find the transfer function of two parallel filters?  Do I
> merely sum the two together?
>
>         sum = H1+H2
>
> I would think this would work, but I would be worried that the phase
> responses would throw the result off.  How do I find the combined
> transfer function?

Yes its the sum of the two.

```
```On 26 Apr, 21:54, Chris Barrett
<"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
> I have two filters in parallel. The transfer functions are given by H1
> and H2. I've illustrated the setup below.
>
>             .-- H1 ---.
>             |         |
> in --------|        (+)------> out
>             |         |
>             '-- H2 ---'
>
> How do I find the transfer function of two parallel filters?  Do I
> merely sum the two together?
>
>         sum = H1+H2
>
> I would think this would work, but I would be worried that the phase
> responses would throw the result off.  How do I find the combined
> transfer function?

The total frequency response is the (complex-valued) sum of the two.
That's a property of linear systems.

Rune

```
```Rune Allnor wrote:
> On 26 Apr, 21:54, Chris Barrett
> <"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
>
>>I have two filters in parallel. The transfer functions are given by H1
>>and H2. I've illustrated the setup below.
>>
>>            .-- H1 ---.
>>            |         |
>>in --------|        (+)------> out
>>            |         |
>>            '-- H2 ---'
>>
>>How do I find the transfer function of two parallel filters?  Do I
>>merely sum the two together?
>>
>>        sum = H1+H2
>>
>>I would think this would work, but I would be worried that the phase
>>responses would throw the result off.  How do I find the combined
>>transfer function?
>
>
> The total frequency response is the (complex-valued) sum of the two.
> That's a property of linear systems.
>
> Rune
>

How would I determine whether the system is linear or non-linear?
```
```On Apr 27, 1:11 pm, Chris Barrett
<"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
> Rune Allnor wrote:
> > On 26 Apr, 21:54, Chris Barrett
> > <"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
>
> >>I have two filters in parallel. The transfer functions are given by H1
> >>and H2. I've illustrated the setup below.
>
> >>            .-- H1 ---.
> >>            |         |
> >>in --------|        (+)------> out
> >>            |         |
> >>            '-- H2 ---'
>
> >>How do I find the transfer function of two parallel filters?  Do I
> >>merely sum the two together?
>
> >>        sum = H1+H2
>
> >>I would think this would work, but I would be worried that the phase
> >>responses would throw the result off.  How do I find the combined
> >>transfer function?
>
> > The total frequency response is the (complex-valued) sum of the two.
> > That's a property of linear systems.
>
> > Rune
>
> How would I determine whether the system is linear or non-linear?- Hide quoted text -
>
> - Show quoted text -

You do need to figure this out. If the system is nonlinear then you
don't have a transfer function, ... right?

Dirk

```
```Chris Barrett wrote:
> Rune Allnor wrote:
>> On 26 Apr, 21:54, Chris Barrett
>> <"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
>>
>>> I have two filters in parallel. The transfer functions are given by H1
>>> and H2. I've illustrated the setup below.
>>>
>>>            .-- H1 ---.
>>>            |         |
>>> in --------|        (+)------> out
>>>            |         |
>>>            '-- H2 ---'
>>>
>>> How do I find the transfer function of two parallel filters?  Do I
>>> merely sum the two together?
>>>
>>>        sum = H1+H2
>>>
>>> I would think this would work, but I would be worried that the phase
>>> responses would throw the result off.  How do I find the combined
>>> transfer function?
>>
>>
>> The total frequency response is the (complex-valued) sum of the two.
>> That's a property of linear systems.
>>
>> Rune
>>
>
> How would I determine whether the system is linear or non-linear?

The existence of H1 and H2 implies linearity. Nonlinear systems don't
have transfer functions per se. Not every input-output relation is a
transfer function.

Jerry
--
Engineering is the art of making what you want from things you can get.
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```