Convert a piecewise definition of a single impulse into an impulse train

Question

NOTE: See update at end of question

I have a function smoothstep (based on the derivative of a smoothstep function) that gives a single impulse as follows:

smoothstep = Piecewise[{{0, x <= -1}, 
       {(1/32)*(35 - 105*x^2 + 105*x^4 - 35*x^6), -1 < x < 1}, 
       {1, x >= 1}}]
Plot[{smoothstep}, {x, -1, 1}, GridLines -> Automatic, 
   PlotLegends -> "Expressions"]

**Piecewise[{{0, x <= -1}, 
     {(1/32)*(35 - 105*x^2 + 105*x^4 - 35*x^6), -1 < x < 1}, 
     {1, x >= 1}}, 0]**

I want to use this as the basis for a train of impulses, where single impulses like the one above appear at all instances of x+n a where n is the set of integers and a is a chosen positive integer. For example, for a=4, the function would return impulses in each of the ranges {...-9<=x<=-7,-5<=x<=-3,,-1<=x<=1,3<=x<=5,7<=x<=9...}, and return 0 in the ranges {...-7<x<-5,-3<x<-1,1<x<3,5<x<7...}.

I can't figure out the syntax to deliver this as a new piecewise function. How do I define n and a? Presumably it involves slots or double-slots...?

(I tried the following with no success:)

smoothstepimpulsetrain = Piecewise[
     {{(1/32)*(35 - 105*(x - n*a)^2 + 105*(x - n*a)^4 - 
              35*(x - n*a)^6), Element[n, Integers]}, 
       {0, NotElement[n, Integers]}}]
Plot[{smoothstepimpulsetrain /. a -> 4}, {x, 15, 15}, 
   GridLines -> Automatic, PlotLegends -> "Expressions"]

UPDATE

@Michael E2 and @MikeY made the following suggestion, but it doesn't quite work, as you can see from the plot for a=1:

smoothstep[x_] := Piecewise[{{0, x <= -1}, 
    {(1/32)*(35 - 105*x^2 + 105*x^4 - 35*x^6), -1 < x < 1}, 
    {0, x >= 1}}];
train[x_, a_] := smoothstep[Mod[x, a, -1]];
Plot[{train[x, 1], train[x, 3], train[x, 5], train[x, 7]}, 
    {x, 0, 10}]

Also note that summing these trains produces an incorrect plot - note the feeble little bump at x=5, and the anomalous behaviour in the range x=6->8:

Plot[{train[x, a] /. a -> 3 + train[x, a] /. 
     a -> 5 + train[x, a] /. a -> 7}, {x, 0, 10}]

Further suggestions?

Answer 1

UPDATED to allow overlapping pulses

Convert to function

smoothstep[x_] := Piecewise[{{0, x <= -1}, {(1/32)*(35 - 105*x^2 + 105*x^4 - 35*x^6), -1 < x <  1}, {0, x >= 1}}]

Modify to make the train, noting that the code assumes the pulse has a width of +/- $d$ and accepts that as input. for your code, $d=1$.

train[x_, a_, d_] := Module[{
                     kl = Floor[(x - d)/a] - 1, 
                     kr = Ceiling[(x + d)/a] + 1,
                     k = 0, 
                     summand = 0
                     },

  For[k = kl, k <= kr, k++, summand += smoothstep[x - k a]];

  summand

   ]

Plot letting a=4

Plot[train[x, 4, 1], {x, 0, 10}]

Plot again with an overlap of pulses ($a=.3$)

Plot[train[x, .3, 1], {x, 1, 3}]